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Prove that $\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\tan 2 x$
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L.H.S. $=\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\frac{\sin 3 x+\sin x}{\cos 3 x+\cos x}$
$=\frac{2 \sin \frac{3 x+x}{2} \cos \frac{3 x-x}{2}}{2 \cos \frac{3 x+x}{2} \cos \frac{3 x-x}{2}}=\frac{\sin 2 x \cos x}{\cos 2 x \cos x}$
$=\frac{\sin 2 x}{\cos 2 x}=\tan 2 x$
$=\frac{2 \sin \frac{3 x+x}{2} \cos \frac{3 x-x}{2}}{2 \cos \frac{3 x+x}{2} \cos \frac{3 x-x}{2}}=\frac{\sin 2 x \cos x}{\cos 2 x \cos x}$
$=\frac{\sin 2 x}{\cos 2 x}=\tan 2 x$
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