Question:
Prove that:
$\frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}=\tan x$
Solution:
$\mathrm{LHS}=\frac{\sin x+\sin 2 x}{1+\cos x+\cos 2 x}$
$=\frac{\sin x+\sin 2 x}{\cos x+(1+\cos 2 x)}$
$=\frac{\sin x+2 \sin x \cos x}{\cos x+2 \cos ^{2} x} \quad\left[\because \sin 2 x=2 \sin x \cos x\right.$ and $\left.2 \cos ^{2} x=1+\cos 2 x\right]$
$=\frac{\sin x(1+2 \cos x)}{\cos x(1+2 \cos x)}$
$=\tan x=\mathrm{RHS}$
Hence proved.
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