Prove the following work the problem either from right left

Prove the following work the problem either from right left like we did in one class. 1 + cos x/1 + sin x + cos x = sin x/1 + sin x - cos x

(1+cosx)/(1 +sinx+cosx) = sinx/( 1+sinx -cosx)

LHS : (1+cosx)/(1 +sinx+cosx)

= (1 +2cos^2x/2 -1)/( 1 +2sinx/2cosx/2 + 2cos^2x/2 -1)

= 2cos^2(x/2) /( 2cos(x/2)[ sinx/2 + cosx/2]

= cos(x/2)/ (sin(x/2) +cos(x/2)

Now simplify RHS

RHS : sinx/( 1+sinx -cosx)

= 2sinx/2cosx/2 /( 1+2sinx/2cosx/2 - (1- 2sin^2(x/2) )

= 2sin(x/2)cos(x/2)/( 2sin^2(x/2) + 2sinx/2cosx/2)

= cos(x/2)/(sinx/2 +cosx/2)

Hence RHS = LHS