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Trigonometric Identities for Angle Products

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Theorem: Chebyshev's Formulas

The sine, cosine and tangent obey Chebyshev’s formulas for every :

\begin{align*}\sin(r\varphi) &= 2\cos \varphi \sin ((r-1)\varphi) - \sin ((r-2)\varphi) \\ \\ \cos (r \varphi) &= 2\cos \varphi \cos ((r-1)\varphi) - \cos((r-2)\varphi) \\ \\ \tan(r\varphi) &= \frac{\tan((r-1)\varphi) + \tan \varphi}{1 - \tan \varphi\tan ((r-1)\varphi)} \end{align*}

Theorem: Double-Angle Formulas

The sine, cosine, tangent and cotangent of can be expressed as

\begin{align*} \sin(2\theta) &= 2\sin \theta \cos \theta \\ \\ \cos(2\theta) &= 2\cos^2\theta + 1 = \cos^2 \theta - \sin^2 \theta = 1-2\sin^2 \theta \\ \\ \tan (2\theta) &= \frac{2\tan \theta}{1-\tan^2 \theta} \end{align*}

Theorem: Half-Angle Formulae

The sine, cosine, tangent and cotangent obey the following properties:

\begin{align*}\sin \frac{\theta}{2} &= \operatorname{sgn}\left(\sin \frac{\theta}{2}\right) \sqrt{\frac{1-\cos\theta}{2}} \\ \\ \cos \frac{\theta}{2} &= \operatorname{sgn}\left(\cos \frac{\theta}{2}\right) \sqrt{\frac{1+\cos\theta}{2}} \\ \\ \tan\frac{\theta}{2} &= \frac{\sin \theta}{1+\cos \theta} = \frac{1 - \cos \theta}{\sin \theta} = \operatorname{sgn}(\sin \theta) \sqrt{\frac{1-\cos \theta}{1+\cos \theta}} \\ \\ \cot \frac{\theta}{2} &= \frac{1+\cos \theta}{\sin \theta} = \frac{\sin \theta}{1 - \cos \theta} = \pm \sqrt{\frac{1+\cos \theta}{1-\cos \theta}} \end{align*}

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