Question

Prove that:
$\frac{\sin x+\sin 2x}{1+\cos x+\cos 2x}=\tan x$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\frac{\sin x+\sin 2x}{1+\cos x+\cos 2x} \\ =\frac{\sin x+\sin 2x}{\cos x+\left(1+\cos 2x\right)} \end{gathered}$$

$=\frac{\sin x+2\sin x\cos x}{\cos x+2{\cos }^{2}x}\left[\because \sin 2x=2\sin x\cos x\mathrm{and}2{\cos }^{2}x=1+\cos 2x\right]$

$$\begin{gathered} =\frac{\sin x\left(1+2\cos x\right)}{\cos x\left(1+2\cos x\right)} \\ =\tan x=\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$