Rewrite $\sin^4 x$ so that it involves only the first power of cosine? What I have so far: 
rewrite the problem as a product of two squared terms: 
$\sin^2(x)\sin^2(x)$ 
Apply the squared identity for sine twice: 
$(1+\cos(x)) (1-\cos(x)) (1+\cos(x)) (1-\cos(x))$ 
Multiply the terms: 
$(1+\cos(x))^2 (1-\cos(x))^2$ 
Then they want me to apply the squared identity for cosine and then simplify. 
How do I apply the identity when it is squared? 
Thanks.

Good try! I suggest using this identity instead:
$$\sin^2 x = \frac{1}{2} \left(1-\cos 2 x\right).$$
That one will get you halfway. Next you'll need:
$$\cos^2 x = \frac{1}{2} \left(1+\cos 2 x\right).$$
