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Math Help - show by algebraic means the value of sin^4(15) - cos^4(15_

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    show 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
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    Re: show by algebraic means the value of sin^4(15) - cos^4(15_

    Quote Originally Posted by koolaid123 View Post
    i know they are difference of squares but i dont know what to do after....the answer is -root3over2
    \begin{align*} \sin^4(15^o)- \cos^4 (15^o) &= [\sin^2(15^o)- \cos^2 (15^o)][ \sin^2(15^o)+\cos^2 (15^o)] \\   &=\sin^2(15^o)- \cos^2 (15^o)  \\ &= -\cos(30^o)\end{align*}
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    MHF Contributor MarkFL's Avatar
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    Re: show by algebraic means the value of sin^4(15) - cos^4(15_

    We are given to evaluate:

    \sin^4(15^{\circ})-\cos^4(15^{\circ})

    You are right, recognizing this as a difference of squares is a good start:

    (\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:

    -(\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...
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