# Cos X + Cos 3X + Cos 5X Cos 7X Sin X + Sin 3X Sin 5X Sin 7X Cot 4X

$$\frac{sin\,x-sin\,3x+sin\,5x-sin\,7x}{cos\,x-cos\,3x-cos\,5x+cos\,7x}=cot\,2x$$

sin x- sin 3x+sin 5x-sin 7x/ co x-cos 3x-cos 5x+cos 7x = cot 2x

L.H.S =$$\frac{sin\,x-sin\,3x+sin\,5x-sin\,7x}{cos\,x-cos\,3x-cos\,5x+cos\,7x}$$

=$$\frac{(5sin\,x + sin\,x)-(sin\,7x+sin\,3x)}{(cos\,x-cos5x)-(cos\,3x-cos\,7x)}$$

=$$\frac{2sin(\frac{5x+x}{2}).cos(\frac{5x-x}{2})-2sin(\frac{7x+3x}{2}).cos(\frac{7x-3x}{2})}{2sin(\frac{x+5x)}{2}).sin(\frac{5x-x}{2})-2sin(\frac{3x+7x}{2}).sin(\frac{7x-3x}{2})}$$

=$$\frac{2\,sin\,3x\,.cos\,2x-2\,sin\,5x.cos\,2x}{2\,sin\,3x.sin\,2x-2sin\,5x.sin\,2x}$$

=$$\frac{2\,cos\,2x(sin\,3x-sin\,5x)}{2\,sin\,2x(sin\,3x-sin\,\,5x)}$$

=$$\frac{cos\,2x}{sin\,2x}=cot\, 2x = R.H.S.$$

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+1 vote
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