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Subsection B.2.6 Identities — Double-angle Formulas

If we set \(\beta=\alpha\) in the angle-addition formulas we get

\begin{align*} \sin(2\alpha) &= 2\sin(\alpha)\cos(\alpha)\\ \cos(2\alpha) &= \cos^2(\alpha)-\sin^2(\alpha)\\ &= 2\cos^2(\alpha)-1 & \text{since } \sin^2\theta =1-\cos^2\theta\\ &= 1-2\sin^2(\alpha) & \text{since } \cos^2\theta =1-\sin^2\theta\\ \tan(2\alpha) &= \frac{2\tan(\alpha)}{1-\tan^2(\alpha)}\\ &= \frac{2}{\cot(\alpha)-\tan(\alpha)} &\text{divide top and bottom by $\tan(\alpha)$} \end{align*}