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| '''Proofs of trigonometric identities''' are used to show relations between [[trigonometric functions]]. This article will list trigonometric identities and prove them.
| | The name of the writer is Numbers. Managing people is what I do and the wage has been truly satisfying. North Dakota is where me and my husband reside. Body building is 1 of the issues I love most.<br><br>Feel free to visit my web-site ... home std test ([http://bikedance.com/blogs/post/15246 Full Piece of writing]) |
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| ==Elementary trigonometric identities==
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| ===Definitions===
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| [[Image:Trigonometric Triangle.svg|right|thumb|Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.|396x396px]]
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| Referring to the diagram at the right, the six trigonometric functions of θ are:
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| :<math> \sin \theta = \frac {\mathrm{opposite}}{\mathrm{hypotenuse}} = \frac {a}{h}</math>
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| :<math> \cos \theta = \frac {\mathrm{adjacent}}{\mathrm{hypotenuse}} = \frac {b}{h}</math>
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| :<math> \tan \theta = \frac {\mathrm{opposite}}{\mathrm{adjacent}} = \frac {a}{b}</math>
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| :<math> \cot \theta = \frac {\mathrm{adjacent}}{\mathrm{opposite}} = \frac {b}{a}</math>
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| :<math> \sec \theta = \frac {\mathrm{hypotenuse}}{\mathrm{adjacent}} = \frac {h}{b}</math>
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| :<math> \csc \theta = \frac {\mathrm{hypotenuse}}{\mathrm{opposite}} = \frac {h}{a}</math>
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| ===Ratio identities===
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| The following identities are trivial algebraic consequences of these definitions and the division identity.<br> | |
| c is whatever value (not necessarily trigonometric), only to understand the simple demonstrations above.
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| That is because not appear in the graph.
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| : <math> \frac {a}{b}= \frac {\left(\frac {a}{c}\right)} {\left(\frac {b}{c}\right) }.</math>
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| :<math> \tan \theta
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| = \frac{\mathrm{opposite}}{\mathrm{adjacent}}
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| = \frac { \left( \frac{\mathrm{opposite}}{\mathrm{hypotenuse}} \right) } { \left( \frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\right) }
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| = \frac {\sin \theta} {\cos \theta}. </math>
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| :<math> \cot \theta = \frac {\cos \theta}{\sin \theta}.</math>
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| :<math> \cot \theta =\frac{\mathrm{adjacent}}{\mathrm{opposite}}
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| = \frac { \left( \frac{\mathrm{adjacent}}{\mathrm{adjacent}} \right) } { \left( \frac {\mathrm{opposite}}{\mathrm{adjacent}} \right) } = \frac {1}{\tan \theta}.</math>
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| :<math> \sec \theta = \frac {1}{\cos \theta}</math>
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| :<math> \csc \theta = \frac {1}{\sin \theta}</math>
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| :<math> \tan \theta = \frac{\mathrm{opposite}}{\mathrm{adjacent}}
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| = \frac{\left(\frac{\mathrm{opposite} \times \mathrm{hypotenuse}}{\mathrm{opposite} \times \mathrm{adjacent}} \right) } { \left( \frac {\mathrm{adjacent} \times \mathrm{hypotenuse}} {\mathrm{opposite} \times \mathrm{adjacent} } \right) }
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| = \frac{\left( \frac{\mathrm{hypotenuse}}{\mathrm{adjacent}} \right)} { \left( \frac{\mathrm{hypotenuse}}{\mathrm{opposite}} \right)}
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| = \frac {\sec \theta}{\csc \theta}. </math>
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| :<math> \cot \theta = \frac {\csc \theta}{\sec \theta}.</math>
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| ===Complementary angle identities===
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| Two angles whose sum is π/2 radians (90 degrees) are ''complementary''. In the diagram, the angles at vertices A and B are complementary, so we can exchange a and b, and change θ to π/2 − θ, obtaining:
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| :<math> \sin\left( \pi/2-\theta\right) = \cos \theta</math>
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| :<math> \cos\left( \pi/2-\theta\right) = \sin \theta</math>
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| :<math> \tan\left( \pi/2-\theta\right) = \cot \theta</math>
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| :<math> \cot\left( \pi/2-\theta\right) = \tan \theta</math>
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| :<math> \sec\left( \pi/2-\theta\right) = \csc \theta</math>
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| :<math> \csc\left( \pi/2-\theta\right) = \sec \theta</math>
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| === Pythagorean identities ===
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| Identity 1:
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| :<math>\sin^2(x) + \cos^2(x) = 1\,</math>
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| Proof 1:
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| Refer to the triangle diagram above. Note that <math>a^2+b^2=h^2</math> by [[Pythagorean theorem]].
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| :<math>\sin^2(x) + \cos^2(x) = \frac{a^2}{h^2} + \frac{b^2}{h^2} = \frac{a^2+b^2}{h^2} = \frac{h^2}{h^2} = 1.\, </math>
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| The following two results follow from this and the ratio identities. To obtain the first, divide both sides of <math>\sin^2(x) + \cos^2(x) = 1</math> by <math>\cos^2(x)</math>; for the second, divide by <math>\sin^2(x)</math>.
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| :<math>\tan^2(x) + 1\ = \sec^2(x) </math>
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| :<math>\sec^2(x) - \tan^2(x) = 1\ </math>
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| Similarly
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| :<math> 1\ + \cot^2(x) = \csc^2(x) </math>
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| :<math>\csc^2(x) - \cot^2(x) = 1\ </math>
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| Proof 2:
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| Differentiating the left-hand side of the identity yields:
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| :<math>2 \sin x \cdot \cos x - 2 \sin x \cdot \cos x = 0 </math>
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| Integrating this shows that the original identity is equal to a constant, and this constant can be found by plugging in any arbitrary value of x.
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| Identity 2:
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| The following accounts for all three reciprocal functions.
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| :<math> \csc^2(x) + \sec^2(x) - \cot^2(x) = 2\ + \tan^2(x) </math>
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| Proof 1:
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| Refer to the triangle diagram above. Note that <math>a^2+b^2=h^2</math> by [[Pythagorean theorem]].
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| :<math>\csc^2(x) + \sec^2(x) = \frac{h^2}{a^2} + \frac{h^2}{b^2} = \frac{a^2+b^2}{a^2} + \frac{a^2+b^2}{b^2} = 2\ + \frac{b^2}{a^2} + \frac{a^2}{b^2}</math>
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| Substituting with appropriate functions -
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| :<math> 2\ + \frac{b^2}{a^2} + \frac{a^2}{b^2} = 2\ + \tan^2(x)+ \cot^2(x) </math>
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| Rearranging gives:
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| :<math> \csc^2(x) + \sec^2(x) - \cot^2(x) = 2\ + \tan^2(x) </math>
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| ===Angle sum identities===
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| ====Sine====
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| [[Image:TrigSumFormula.svg|right|thumb|350px|Illustration of the sum formula.]]
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| Draw the angles α and β. Place P on the line defined by α + β at unit distance from the origin.
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| Let PQ be a perpendicular from P to the line defined by the angle α.
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| OQP is a right angle.
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| Let QA be a perpendicular from Q to the x axis, and PB be a perpendicular from P to the x axis.
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| OAQ is a right angle.
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| Draw QR parallel to the ''x''-axis.
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| Now angle RPQ = α (because OQA = 90 - α, making RQO = α, RQP = 90-α , and finally RPQ = α )
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| <math>RPQ = \tfrac{\pi}{2} - RQP = \tfrac{\pi}{2} - (\tfrac{\pi}{2} - RQO) = RQO = \alpha</math>
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| :<math>OP = 1\,</math>
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| :<math>PQ = \sin \beta\,</math>
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| :<math>OQ = \cos \beta\,</math>
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| :<math>\frac{AQ}{OQ} = \sin \alpha\,</math>, so <math>AQ = \sin \alpha \cos \beta\,</math>
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| :<math>\frac{PR}{PQ} = \cos \alpha\,</math>, so <math>PR = \cos \alpha \sin \beta\,</math>
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| :<math>\sin (\alpha + \beta) = PB = RB+PR = AQ+PR = \sin \alpha \cos \beta + \cos \alpha \sin \beta\,</math>
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| By substituting <math>-\beta</math> for <math>\beta</math> and using [[List of trigonometric identities#Symmetry|Symmetry]], we also get:
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| :<math>\sin (\alpha - \beta) = \sin \alpha \cos -\beta + \cos \alpha \sin -\beta\,</math>
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| :<math>\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta\,</math>
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| Another simple "proof" can be given using Euler's formula known from complex analysis:
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| Euler's formula is:
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| :<math>e^{i\varphi}=\cos \varphi +i \sin \varphi</math>
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| Although it is more precise to say that Euler's formula entails the trigonometric identities, it follows that for angles <math>\alpha</math> and <math>\beta</math> we have:
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| :<math>e^{i (\alpha + \beta)} = \cos (\alpha +\beta) + i \sin(\alpha +\beta)</math>
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| Also using the following properties of exponential functions:
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| :<math>e^{i(\alpha + \beta)} = e^{i \alpha} e^{i\beta}= (\cos \alpha +i \sin \alpha) (\cos \beta + i \sin \beta)</math>
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| Evaluating the product:
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| :<math>e^{i(\alpha + \beta)} = (\cos \alpha \cos \beta - \sin \alpha \sin \beta)+i(\sin \alpha \cos \beta + \sin \beta \cos \alpha)</math>
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| This will only be equal to the previous expression we got, if the imaginary and real parts are equal respectively.
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| Hence we get:
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| :<math>\cos (\alpha +\beta)=\cos \alpha \cos \beta - \sin \alpha \sin \beta</math>
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| :<math>\sin (\alpha +\beta)=\sin \alpha \cos \beta + \sin \beta \cos \alpha</math>
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| ====Cosine====
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| Using the figure above,
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| :<math>OP = 1\,</math>
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| :<math>PQ = \sin \beta\,</math>
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| :<math>OQ = \cos \beta\,</math>
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| :<math>\frac{OA}{OQ} = \cos \alpha\,</math>, so <math>OA = \cos \alpha \cos \beta\,</math>
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| :<math>\frac{RQ}{PQ} = \sin \alpha\,</math>, so <math>RQ = \sin \alpha \sin \beta\,</math>
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| :<math>\cos (\alpha + \beta) = OB = OA-BA = OA-RQ = \cos \alpha \cos \beta\ - \sin \alpha \sin \beta\,</math>
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| By substituting <math>-\beta</math> for <math>\beta</math> and using [[List of trigonometric identities#Symmetry|Symmetry]], we also get:
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| :<math>\cos (\alpha - \beta) = \cos \alpha \cos - \beta\ - \sin \alpha \sin - \beta\,</math>
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| :<math>\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta\,</math>
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| Also, using the complementary angle formulae,
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| :<math>\cos (\alpha + \beta) = \sin\left( \pi/2-(\alpha + \beta)\right) = \sin\left( (\pi/2-\alpha) - \beta\right)\,</math>
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| ::<math>= \sin\left( \pi/2-\alpha\right) \cos \beta - \cos\left( \pi/2-\alpha\right) \sin \beta\,</math>
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| ::<math>= \cos \alpha \cos \beta - \sin \alpha \sin \beta\,</math>
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| ====Tangent and cotangent====
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| From the sine and cosine formulae, we get
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| :<math>\tan (\alpha + \beta) = \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)}\,</math>
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| :<math>= \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta}\,</math>
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| Dividing both numerator and denominator by cos α cos β, we get
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| :<math>\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\,</math>
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| :<math>\tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}\,</math>
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| Similarly from the sine and cosine formulae, we get
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| :<math>\cot (\alpha + \beta) = \frac{\cos (\alpha + \beta)}{\sin (\alpha + \beta)}\,</math>
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| :<math>= \frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\sin \alpha \cos \beta + \cos \alpha \sin \beta}\,</math>
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| Then by dividing both numerator and denominator by sin α sin β, we get
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| :<math>\cot (\alpha + \beta) = \frac{\cot \alpha \cot \beta - 1}{\cot \alpha + \cot \beta}\,</math>
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| :<math>\cot (\alpha - \beta) = \frac{\cot \alpha \cot \beta + 1}{\cot \beta - \cot \alpha}\,</math>
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| === Double-angle identities ===
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| From the angle sum identities, we get
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| :<math>\sin (2 \theta) = 2 \sin \theta \cos \theta\,</math>
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| and
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| :<math>\cos (2 \theta) = \cos^2 \theta - \sin^2 \theta\,</math>
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| The Pythagorean identities give the two alternative forms for the latter of these:
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| :<math>\cos (2 \theta) = 2 \cos^2 \theta - 1\,</math>
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| :<math>\cos (2 \theta) = 1 - 2 \sin^2 \theta\,</math>
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| The angle sum identities also give
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| :<math>\tan (2 \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} = \frac{2}{\cot \theta - \tan \theta}\,</math>
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| :<math>\cot (2 \theta) = \frac{\cot^2 \theta - 1}{2 \cot \theta} = \frac{\cot \theta - \tan \theta}{2}\,</math>
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| It can also be proved using [[Euler's formula]]
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| :<math> e^{i \varphi}=\cos \varphi +i \sin \varphi</math>
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| Squaring both sides yields
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| :<math> e^{i 2\varphi}=(\cos \varphi +i \sin \varphi)^{2}</math>
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| But replacing the angle with its doubled version, which achieves the same result in the left side of the equation, yields
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| :<math> e^{i 2\varphi}=\cos 2\varphi +i \sin 2\varphi</math>
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| It follows that
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| :<math>(\cos \varphi +i \sin \varphi)^{2}=\cos 2\varphi +i \sin 2\varphi</math>.
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| Expanding the square and simplifying on the left hand side of the equation gives
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| :<math>i(2 \sin \varphi \cos \varphi) + \cos^2 \varphi - \sin^2 \varphi\ = \cos 2\varphi +i \sin 2\varphi</math>.
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| Because the imaginary and real parts have to be the same, we are left with the original identities
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| :<math>\cos^2 \varphi - \sin^2 \varphi\ = \cos 2\varphi</math>,
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| and also
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| :<math>2 \sin \varphi \cos \varphi = \sin 2\varphi</math>.
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| === Half-angle identities ===
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| The two identities giving the alternative forms for cos 2θ lead to the following equations:
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| :<math>\cos \frac{\theta}{2} = \pm\, \sqrt\frac{1 + \cos \theta}{2},\,</math>
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| :<math>\sin \frac{\theta}{2} = \pm\, \sqrt\frac{1 - \cos \theta}{2}.\,</math>
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| The sign of the square root needs to be chosen properly—note that if 2π is added to θ, the quantities inside the square roots are unchanged, but the left-hand-sides of the equations change sign. Therefore the correct sign to use depends on the value of θ.
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| For the tan function, the equation is:
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| :<math>\tan \frac{\theta}{2} = \pm\, \sqrt\frac{1 - \cos \theta}{1 + \cos \theta}.\,</math>
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| Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to:
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| :<math>\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}.\,</math>
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| Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
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| :<math>\tan \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta}.\,</math>
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| This also gives:
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| :<math>\tan \frac{\theta}{2} = \csc \theta - \cot \theta.\,</math>
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| Similar manipulations for the cot function give:
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| :<math>\cot \frac{\theta}{2} = \pm\, \sqrt\frac{1 + \cos \theta}{1 - \cos \theta} = \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin \theta}{1 - \cos \theta} = \csc \theta + \cot \theta.\,</math>
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| ===Miscellaneous -- the triple tangent identity===
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| :<math>\text{If }\psi + \theta + \phi = \pi = \text{half circle,}\, </math>
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| ::<math>\text{then }\tan(\psi) + \tan(\theta) + \tan(\phi) = \tan(\psi)\tan(\theta)\tan(\phi).\,</math>
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| Proof:<ref>http://mathlaoshi.com/tags/tangent-identity/</ref>
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| ::<math>\psi = \pi - \theta - \phi\,</math>
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| ::<math>\tan(\psi) = \tan(\pi - \theta - \phi) = - \tan(\theta + \phi) = \frac{- \tan\theta - \tan\phi}{1 - \tan\theta \tan\phi}</math>
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| So
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| ::<math>(1 - \tan\theta \tan\phi) \tan\psi + \tan\theta + \tan\phi = 0\,</math>
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| So
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| ::<math>\tan\psi - \tan\theta \tan\phi \tan\psi + \tan\theta + \tan\phi = 0\,</math>
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| ===Miscellaneous -- the triple cotangent identity===
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| :<math>\text{If }\psi + \theta + \phi = \tfrac{\pi}{2} = \text{quarter circle,}\, </math>
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| ::<math>\text{then }\cot(\psi) + \cot(\theta) + \cot(\phi) = \cot(\psi)\cot(\theta)\cot(\phi).\,</math>
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| Proof:
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| Replace each of <math>\psi\, </math>, <math>\theta\, </math>, and <math>\phi\, </math> with their complementary angles, so cotangents turn into tangents and vice-versa.
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| Now if
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| ::<math>\psi + \theta + \phi = \tfrac{\pi}{2}\, </math>
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| then
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| ::<math>(\tfrac{\pi}{2}-\psi) + (\tfrac{\pi}{2}-\theta) + (\tfrac{\pi}{2}-\phi) = \tfrac{3\pi}{2} - (\psi+\theta+\phi) = \tfrac{3\pi}{2} - \tfrac{\pi}{2} = \pi\, </math>
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| so the result follows from the triple tangent identity.
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| === Prosthaphaeresis identities ===
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| * <math>\sin \theta \pm \sin y = 2 \sin \frac{\theta\pm y}2 \cos \frac{\theta\mp y}2</math>
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| * <math>\cos \theta + \cos y = 2 \cos \frac{\theta+y}2 \cos \frac{\theta-y}2</math>
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| * <math>\cos \theta - \cos y = -2 \sin \frac{\theta+y}2 \sin \frac{\theta-y}2</math>
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| <!-- these need to be proven -->
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| === Inequalities ===
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| [[Image:TrigInequality.svg|right|thumb|342px|Illustration of the sine and tangent inequalities.]]
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| The figure at the right shows a sector of a circle with radius 1. The sector is θ/(2π) of the whole circle, so its area is θ/2.
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| :<math>OA = OD = 1\,</math>
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| :<math>AB = \sin \theta\,</math>
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| :<math>CD = \tan \theta\,</math>
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| The area of triangle OAD is AB/2, or sinθ/2. The area of triangle OCD is CD/2, or tanθ/2.
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| Since triangle OAD lies completely inside the sector, which in turn lies completely inside triangle OCD, we have
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| :<math>\sin \theta < \theta < \tan \theta\,</math>
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| This geometric argument applies if 0<θ<π/2. It relies on definitions of [[arc length]] and
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| [[Jordan measure|area]], which act as assumptions, so it is rather a condition imposed in construction of [[trigonometric functions]] than
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| a provable property.<ref>
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| {{cite journal|last=Richman|first=Fred|title=A Circular Argument|journal=The College Mathematics Journal|date=March 1993|volume=24|issue=2|pages=160–162|url=http://www.jstor.org/stable/2686787 .|accessdate=3 November 2012}}</ref> For the sine function, we can handle other values. If θ>π/2, then θ>1. But sinθ≤1 (because of the Pythagorean identity), so sinθ<θ. So we have
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| :<math>\frac{\sin \theta}{\theta} < 1\ \ \ \mathrm{if}\ \ \ 0 < \theta\,</math>
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| For negative values of θ we have, by symmetry of the sine function
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| :<math>\frac{\sin \theta}{\theta} = \frac{\sin (-\theta)}{-\theta} < 1\,</math>
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| Hence
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| :<math>\frac{\sin \theta}{\theta} < 1\ \ \ \mathrm{if}\ \ \ \theta \ne 0\,</math>
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| :<math>\frac{\tan \theta}{\theta} > 1\ \ \ \mathrm{if}\ \ \ 0 < \theta < \frac{\pi}{2}\,</math>
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| ==Identities involving calculus==
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| ===Preliminaries===
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| :<math>\lim_{\theta \to 0}{\sin \theta} = 0\,</math>
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| :<math>\lim_{\theta \to 0}{\cos \theta} = 1\,</math>
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| These can be seen from looking at the diagrams.
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| ===Sine and angle ratio identity===
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| :<math>\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1</math>
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| Proof: From the previous inequalities, we have, for small angles
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| :<math>\sin \theta < \theta < \tan \theta\,</math>, so
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| :<math>\frac{\sin \theta}{\theta} < 1 < \frac{\tan \theta}{\theta}\,</math>, so
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| :<math>\frac{\sin \theta}{\theta \cos \theta} > 1\,</math>, or
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| :<math>\frac{\sin \theta}{\theta} > \cos \theta\,</math>, so
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| :<math>\cos \theta < \frac{\sin \theta}{\theta} < 1\,</math>, but
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| :<math>\lim_{\theta \to 0}{\cos \theta} = 1\,</math>, so
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| :<math>\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}} = 1</math>
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| === Cosine and angle ratio identity ===
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| :<math>\lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta} = 0\,</math>
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| Proof:
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| :<math>\frac{1 - \cos \theta}{\theta} = \frac{1 - \cos^2 \theta}{\theta (1 + \cos \theta)}\,</math>
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| :<math>= \frac{\sin^2 \theta}{\theta (1 + \cos \theta)}\,</math>
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| :<math>= \frac{\sin \theta}{\theta} \times \sin \theta \times \frac{1}{1 + \cos \theta}.\,</math>
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| The limits of those three quantities are 1, 0, and 1/2, so the resultant limit is zero.
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| === Cosine and square of angle ratio identity ===
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| :<math> \lim_{\theta \to 0}\frac{1 - \cos \theta}{\theta^2} = \frac{1}{2} </math>
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| Proof:
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| As in the preceding proof,
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| :<math>\frac{1 - \cos \theta}{\theta^2} = \frac{\sin \theta}{\theta} \times \frac{\sin \theta}{\theta} \times \frac{1}{1 + \cos \theta}.\,</math>
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| The limits of those three quantities are 1, 1, and 1/2, so the resultant limit is 1/2.
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| === Proof of Compositions of trig and inverse trig functions ===
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| All these functions follow from the Pythagorean trigonometric identity. We can prove for instance the function
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| :<math>\sin[\arctan(x)]=\frac{x}{\sqrt{1+x^2}}</math>
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| Proof:
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| We start from
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| :<math>\sin^2\theta+\cos^2\theta=1</math>
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| Then we divide this equation by <math>\cos^2\theta</math>
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| :<math>\cos^2\theta=\frac{1}{\tan^2\theta+1}</math>
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| Then use the substitution <math>\theta=\arctan(x)</math>, also use the Pythagorean trigonometric identity:
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| :<math>1-\sin^2[\arctan(x)]=\frac{1}{\tan^2[\arctan(x)]+1}</math>
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| Then we use the identity <math>\tan[\arctan(x)]\equiv x</math>
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| :<math>\sin[\arctan(x)]=\frac{x}{\sqrt{x^2+1}}</math>
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| == See also ==
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| <div class="references" style="-moz-column-count:2; column-count:2;">
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| * [[List of trigonometric identities]]
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| * [[Bhaskara I's sine approximation formula]]
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| * [[Generating trigonometric tables]]
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| * [[Aryabhata's sine table]]
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| * [[Madhava's sine table]]
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| * [[Table of Newtonian series]]
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| * [[Madhava series]]
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| * [[Unit vector]] (explains direction cosines)
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| * [[Euler's formula]]
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| </div>
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| ==References==
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| * E. T. Whittaker and G. N. Watson. ''A course of modern analysis'', Cambridge University Press, 1952
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| <references />
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| {{DEFAULTSORT:Trigonometric identities, Proofs of}}
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| [[Category:Trigonometry]]
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| [[Category:Article proofs]]
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