(cos^2x+1+sinx)/(cos^2x+3)=(1+sinx)/(2+sinx)

Question

I'm asked to verify this identity. I've been going at it for a hour now with no successful results. 

Answer 1

If you want to prove from the left to the right, here is one way:
(cos^2x+1+sinx)/(cos^2x+3)

= (1-sin^2x + 1+sinx)/(1-sin^2x+3)

= (1+sinx)(1-sinx+1)/[(2+sinx)(2-sinx)]

= (1+sinx)/(2+sinx), after canceling 2-sinx.

Answer 2

(1+sin(x))/(2+sin(x))=(1+sin(x))*(2-sin(x))/[(2+sin(x))*(2-sin(x))]=(2+sin(x)-sin²(x))/(4-sin²(x))=

=[2+sin(x)-(1-cos²(x))]/[4-(1-cos²(x))]=(cos²(x)+1+sin(x))/(cos²(x)+3)