Question

Only a genius can solve it !!


`( \cos(A) )/(1 - \sin(A) ) + ( \sin(A) )/(1 - \cos(A) ) + 1 = ( \sin(A) \cos(A) )/((1 - \sin(A)(1 - \cos(A))`
​

Answer 1

`\text{L.H.S}\\\\=(\cos A)/(1-\sin A) + (\sin A)/(1-\cos A) +1\\\\\\=(\cos A(1-\cos A) + \sin A(1-\sin A) + (1-\sin A)(1 - \cos A))/((1-\sin A)(1 -\cos A))\\\\\\=(\cos A - \cos^2 A + \sin A - \sin^2 A + 1 - \cos A - \sin A + \sin A \cos A)/((1 -\sin A)(1 - \cos A))\\\\\\=(-(\sin^2 A + \cos^2A) +1 + \sin A \cos A)/((1 - \sin A)(1 - \cos A))\\\\\\=(-1 + 1 + \sin A \cos A )/((1 - \sin A)((1 - \cos A))\\\\\\=(0+ \sin A \cos A)/((1 - \sin A)(1 - \cos A))\\`

`=(\sin A \cos A)/((1 - \sin A)(1 - \cos A))\\\\\\=\text{R.H.S}\\\\\text{Proved.}`