how to integrate this?

Question

cos^2 x / cosec^2 x dx

Answer 1

∫cos²(x) dx / csc²(x) =
∫cos²(x) sin²(x) dx =
∫(1/4) sin²(2x) dx =
∫(1/4) (1-cos²(2x)) dx =
∫(1/4) (1-(cos(4x)+1)/2) dx =
∫(-1/8) (cos(4x)-1) dx =
x/8 - sin(4x)/32 + C

Answer 2

Contrary to what others may have you believe, there is such a thing as a cosecant: 1/sin x. Putting things in terms of sin and cos is always a good idea in my book, so if we do that we get cos^2 x * sin^2 x dx. Here, I'd use the handy identity cos^2 x + sin^2 x = 1, or in this case, 1- cos^2 x = sin^2 x. Then, you're left with cos^2 x(1 - cos^2 x) dx = cos^2 x - cos^4 x dx, which can thankfully be split up. From here it shouldn't be too difficult to solve.