How do I simplyfy (Sin(x/2))^2 and then derivate the answer?

Question

So my question is how do i solve f(x) = (sin(x/2))^2 dx

Thank you very much, I have a math exam in two weeks, and I'm trying to do a earlier exam.

Answer 1

f(x) = x^2

g(x) = sin (x/2)

f(g(x)) = (sin(x/2))^2 <- your problem, and you're asked to find (f(g(x)))'

Use the chain rule:  (f(g(x)))' = f'(g(x))g'(x)

f'(x) = 2x; g'(x) = :using the chain rule again: .5cos(x/2)

plugging into the equation; answer:

2(sin(x/2)) * .5cos(x/2) = final answer -> sin(x/2)*cos(x/2) <- final answer

Answer 2

You have set this up like you want the integral, but have asked for the derivative

∫sin²(x/2) dx

= ½ ∫1 - cos(x) dx

= ½ (x - sin(x)) + C

I just used the identity:

sin²(x) = ½ (1 - cos(2x))