cos(3x)=cos^3x-3sin^2xcosx?

Need to prove the identity for homework please help.

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  • Math
    Lv 7
    6 years ago
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    cos 3x = cos^3x-3sin^2x cos x [given]

    PROOF

    ///////////

    Evaluating LHS,

    cos(x+y) = cos(x)cos(y) - sin(x)sin(y) [identity]

    cos(3x) = cos(x + 2x)= cos(x)cos(2x) -sin(x) sin(2x) [manipulation of above identity]

    Manipulation Cos(2x) and sin(2x) using the above identity, we get

    =cos(x)(cos^2(x)-sin^2(x)) -sin(x)(2sin(x)cos(x))

    =cos^3(x) -sin^2(x)cos(x) - 2sin^2(x)cos(x)

    =cos^3x - 3sin^2(x)cos(x)

    Therefore LHS = RHS

    Hence proven

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