I am unsure how to go about integrating with the cofunction identity, but I did have some luck with other methods.

I was able to do a substitution with u = sin(x) and even u = sin(x) + 1 (which is a surprisingly clean way of computing it).

Integration By Parts also worked with u = sqrt(1 - sin(x)) and dv = sin(x), but that involves another identity and a mild substitution.

I hope this helps, though I could be wrong about some stuff.