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

If you want to prove from the left to the right, here is one way:

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

(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.