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Prove that $\frac{\cos 4 x+\cos 3 x+\cos 2 x}{\sin 4 x+\sin 3 x+\sin 2 x}=\cot 3 x$
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L.H.S. $=\frac{\cos 4 x+\cos 3 x+\cos 2 x}{\sin 4 x+\sin 3 x+\sin 2 x}$
$=\frac{\cos 4 x+\cos 2 x+\cos 3 x}{\sin 4 x+\sin 2 x+\sin 3 x}$
$=\frac{2 \cos \frac{4 x+2 x}{2} \cos \frac{4 x-2 x}{2}+\cos 3 x}{2 \sin \frac{4 x+2 x}{2} \cos \frac{4 x-2 x}{2}+\sin 3 x}$
$=\frac{2 \cos 3 x \cos x+\cos 3 x}{2 \sin 3 x \cos x+\sin 3 x}$
$=\frac{\cos 3 x(2 \cos x+1)}{\sin 3 x(2 \cos x+1)}$
$=\frac{\cos 3 x}{\sin 3 x}=\cot 3 x=$ R.H.S.
$=\frac{\cos 4 x+\cos 2 x+\cos 3 x}{\sin 4 x+\sin 2 x+\sin 3 x}$
$=\frac{2 \cos \frac{4 x+2 x}{2} \cos \frac{4 x-2 x}{2}+\cos 3 x}{2 \sin \frac{4 x+2 x}{2} \cos \frac{4 x-2 x}{2}+\sin 3 x}$
$=\frac{2 \cos 3 x \cos x+\cos 3 x}{2 \sin 3 x \cos x+\sin 3 x}$
$=\frac{\cos 3 x(2 \cos x+1)}{\sin 3 x(2 \cos x+1)}$
$=\frac{\cos 3 x}{\sin 3 x}=\cot 3 x=$ R.H.S.
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