Question

Prove that:

$\sqrt{\frac{1-\cos 2\theta }{1+\cos 2\theta }}=\tan \theta$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\sqrt{\frac{1-\cos 2\mathrm{\theta }}{1+\cos 2\mathrm{\theta }}} \\ =\sqrt{\frac{2{\sin }^2\mathrm{\theta }}{2{\cos }^2\mathrm{\theta }}}\left[\because 1-\cos 2\mathrm{\theta }=2{\sin }^2\mathrm{\theta }\mathrm{and}1+\cos 2\mathrm{\theta }=2{\cos }^2\mathrm{\theta }\right] \\ =\frac{\mathrm{sin\theta }}{\mathrm{cos\theta }} \\ =\mathrm{tan\theta }=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$