Question

Can anyone please solve this problem

Answer 1

Solution :

$\small{:\implies \sf{\dfrac{cos(x)}{sin(x) + cos(y)} + \dfrac{cos(y)}{sin(y) - cos(x)} = \dfrac{cos(x)}{sin(x) - cos(y)} + \dfrac{cos(y)}{sin(y) + cos(x)}}}$

$\small{:\implies \sf{\dfrac{cos(x)}{sin(x) + cos(y)} - \dfrac{cos(x)}{sin(x) - cos(y)} = \dfrac{cos(y)}{sin(y) + cos(x)} - \dfrac{cos(y)}{sin(y) - cos(x)}}}$

$\small{:\implies \sf{\dfrac{cos(x)[sin(x) - cos(y)] - cos(x)[sin(x) + cos(y)]}{[sin(x) + cos(y)][sin(x) - cos(y)]} =}} \\ \\ \small{\sf{\dfrac{cos(y)[sin(y) - cos(x)] - cos(y)[sin(y) + cos(x)]}{[sin(y) + cos(x)][sin(y) - cos(x)]}}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{[sin(x) - cos(y)] - [sin(x) + cos(y)]}{[sin(x) + cos(y)][sin(x) - cos(y)]}\bigg]}} = \\ \\ \small{\sf{cos(y)\bigg[\dfrac{[sin(y) - cos(x)] - [sin(y) + cos(x)]}{[sin(y) + cos(x)][sin(y) - cos(x)]}\bigg]}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{sin(x) - cos(y) - sin(x) - cos(y)}{sin^{2}(x) - cos^{2}(y)}\bigg]}} = \\ \\ \small{\sf{cos(y)\bigg[\dfrac{sin(y) - cos(x) - sin(y) - cos(x)}{sin^{2}(y) - cos^{2}(x)}\bigg]}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{- 2cos(y)}{sin^{2}(x) - cos^{2}(y)}\bigg] = cos(y)\bigg[\dfrac{- 2cos(x)}{sin^{2}(y) - cos^{2}(x)}\bigg]}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{- 2cos(y)}{[1 - cos^{2}(x)] - [1 - sin^{2}(y)]}\bigg] = cos(y)\bigg[\dfrac{- 2cos(x)}{sin^{2}(y) - cos^{2}(x)}\bigg]}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{- 2cos(y)}{1 - cos^{2}(x) - 1 + sin^{2}(y)}\bigg] = cos(y)\bigg[\dfrac{- 2cos(x)}{sin^{2}(y) - cos^{2}(x)}\bigg]}}$

$\small{:\implies \sf{cos(x)\bigg[\dfrac{- 2cos(y)}{sin^{2}(y) - cos^{2}(x)}\bigg] = cos(y)\bigg[\dfrac{- 2cos(x)}{sin^{2}(y) - cos^{2}(x)}\bigg]}}$

$\small{:\implies \sf{\dfrac{- 2cos(y)cos(x)}{sin^{2}(y) - cos^{2}(x)} = \dfrac{- 2cos(x)cos(y)}{sin^{2}(y) - cos^{2}(x)}}}$

$\underline{\over{\small{:\implies \sf{\dfrac{- 2cos(y)cos(x)}{sin^{2}(y) - cos^{2}(x)} = \dfrac{- 2cos(x)cos(y)}{sin^{2}(y) - cos^{2}(x)}}}}}$

$\boxed{\therefore \small{\sf{\dfrac{cos(x)}{sin(x) + cos(y)} + \dfrac{cos(y)}{sin(y) - cos(x)} = \dfrac{cos(x)}{sin(x) - cos(y)} + \dfrac{cos(y)}{sin(y) + cos(x)}}}}$

Proved!