i know they are difference of squares but i dont know what to do after....the answer is -root3over2

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- Jan 23rd 2013, 12:40 PMkoolaid123show by algebraic means the value of sin^4(15) - cos^4(15_
i know they are difference of squares but i dont know what to do after....the answer is -root3over2

- Jan 23rd 2013, 01:03 PMPlatoRe: show by algebraic means the value of sin^4(15) - cos^4(15_
- Jan 23rd 2013, 01:06 PMMarkFLRe: show by algebraic means the value of sin^4(15) - cos^4(15_
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...