Question

Why does cos^2(x)= 1/2cos(2x)+1/2? I am trying to integrate, but the answer key says to first rewrite the expression like the above. I don't get how to change cos^2(x) into that. Explain?

Answer 1

both sin^2 x and cos^2 x are not easy to integrate.
but we know that
cos (2x)
= cos^2 x - sin^2 x
or
= 2cos^2 x - 1
or
1 - 2sin^2 x

you should have those available to you

so taking
2cos^2 x - 1 = cos (2x)
2cos^2 x = cos (2x) + 1
cos^2 x = (cos(2x) + 1/2
= (1/2)cos(2x) + 1/2

now you should be able to integrate
(1/2)cos(2x) + 1/2 ----> (1/4)sin(2x) + (1/2)x
and thus you have integrated cos^2 x

btw, to integrate sin^2 x, you would use the other formula for cos 2x