prove that

cosx + cos2x +cos3x+cos4x / sinx+sin2x+sin3x+sin4x =cot5/2 x

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prove that
cosx + cos2x +cos3x+cos4x / sinx+sin2x+sin3x+sin4x =cot5/2 x

Ankush Jain · Accepted Answer

To prove: $\frac{\cos x + \cos 2 x + \cos 3 x + \cos 4 x}{\sin x + \sin 2 x + \sin 3 x + \sin 4 x} = \cot \frac{5 x}{2}$ Proof: LHS = $\frac{\cos x + \cos 2 x + \cos 3 x + \cos 4 x}{\sin x + \sin 2 x + \sin 3 x + \sin 4 x} = \frac{\cos x + \cos 4 x + \cos 3 x + \cos 2 x}{\sin x + \sin 4 x + \sin 3 x + \sin 2 x}$ $= \frac{2 \cos \frac{(x + 4 x)}{2} \cos \frac{(x - 4 x)}{2} + 2 \cos \frac{(3 x + 2 x)}{2} \cos \frac{(3 x - 2 x)}{2}}{2 \sin \frac{(x + 4 x)}{2} \cos \frac{(x - 4 x)}{2} + 2 \sin \frac{(3 x + 2 x)}{2} \cos \frac{(3 x - 2 x)}{2}} = \frac{2 \cos \frac{(5 x)}{2} \cos \frac{(- 3 x)}{2} + 2 \cos \frac{(5 x)}{2} \cos \frac{(x)}{2}}{2 \sin \frac{(5 x)}{2} \cos \frac{(- 3 x)}{2} + 2 \sin \frac{(5 x)}{2} \cos \frac{(x)}{2}} = \frac{2 \cos \frac{(5 x)}{2} [\cos \frac{(3 x)}{2} + \cos \frac{(x)}{2}]}{2 \sin \frac{(5 x)}{2} [\cos \frac{(3 x)}{2} + \cos \frac{(x)}{2}]} = \frac{\cos \frac{(5 x)}{2}}{\sin \frac{(5 x)}{2}} = \cot \frac{5 x}{2}$ = RHS [Hence Proved]

prove that cosx + cos2x +cos3x+cos4x / sinx+sin2x+sin3x+sin4x =cot5/2 x

prove that

cosx + cos2x +cos3x+cos4x / sinx+sin2x+sin3x+sin4x =cot5/2 x