L.H.S = \(\frac{sin\,x-sin\,3x+sin\,5x-sin\,7x}{cos\,x-cos\,3x-cos\,5x+cos\,7x}\)
= \(\frac{(5sin\,x + sin\,x)-(sin\,7x+sin\,3x)}{(cos\,x-cos5x)-(cos\,3x-cos\,7x)}\)
= \(\frac{2sin(\frac{5x+x}{2}).cos(\frac{5x-x}{2})-2sin(\frac{7x+3x}{2}).cos(\frac{7x-3x}{2})}{2sin(\frac{x+5x)}{2}).sin(\frac{5x-x}{2})-2sin(\frac{3x+7x}{2}).sin(\frac{7x-3x}{2})}\)
=\(\frac{2\,sin\,3x\,.cos\,2x-2\,sin\,5x.cos\,2x}{2\,sin\,3x.sin\,2x-2sin\,5x.sin\,2x}\)
= \(\frac{2\,cos\,2x(sin\,3x-sin\,5x)}{2\,sin\,2x(sin\,3x-sin\,\,5x)}\)
= \(\frac{cos\,2x}{sin\,2x}=cot\, 2x = R.H.S.\)