Use complex number methods to show that $\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$. H

Question

Use complex number methods to show that $\tan 2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$. Hence find the exact value of $\tan\dfrac{\pi}{8}$. Use real $\theta$ for which $\cos\theta\ne0$ and $\cos2\theta\ne0$, so both sides of the tangent identity are defined.

Accepted Answer

(a) Shown; $\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}$ (b) $\tan\dfrac{\pi}{8}=\sqrt2-1$

Double angle tangent identity via complex numbers

Double angle tangent identity via complex numbers

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