i know they are difference of squares but i dont know what to do after....the answer is -root3over2
We are given to evaluate:
$\displaystyle \sin^4(15^{\circ})-\cos^4(15^{\circ})$
You are right, recognizing this as a difference of squares is a good start:
$\displaystyle (\sin^2(15^{\circ})+\cos^2(15^{\circ}))(\sin^2(15^ {\circ})-\cos^2(15^{\circ}))$
Next, with the angle sum/difference identities for cosine in mind, let's rewrite this as:
$\displaystyle -(\sin^2(15^{\circ})+\cos^2(15^{\circ}))(\cos^2(15^ {\circ})-\sin^2(15^{\circ}))$
The first factor should look familiar as a Pythagorean identity, and the second factor should look familiar as a double-angle identity for cosine...