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Math Help - Trig iden: Quadruple angle formulas

  1. #1
    Junior Member
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    Trig iden: Quadruple angle formulas

    How do you tackle these?
    1. 2cos^2 4x-1
    sin4xcos4x





    2. 1-4sin^2x+4sin^4x

    The answers to the first is 2cot8x and the second is cos^22x. I would like to see the step by step process.
    Help would be appreciated thanks!
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  2. #2
    Senior Member tukeywilliams's Avatar
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    Is it:

    1.  \frac{2 \cos^{2} 4x-1}{\sin 4x \cos 4x} and

    2.  1 - 4 \sin^{2}x + 4 \sin^{4} x ?
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    We use the addition formulas for sine and cosine here.

    Recall:

    \cos 2x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x

    \Rightarrow \cos 8x = 2 \cos^2 4x - 1

    \sin 2x = 2 \sin x \cos x

    \Rightarrow \sin 8x = 2 \sin 4x \cos 4x

    Quote Originally Posted by andrewsx View Post
    How do you tackle these?
    1. 2cos^2 4x-1
    sin4xcos4x
    \frac {2 \cos^2 4x - 1}{\sin 4x \cos 4x}

    = \frac {\cos 8x}{ \frac {1}{2} \sin 8x}

    = 2 \frac {\cos 8x }{\sin 8x}

    = 2 \cot 8x


    2. 1-4sin^2x+4sin^4x
    1 - 4 \sin^2 x + 4 \sin^4 x ............Note that this is quadratic in \sin^2 x, so let's foil

    = \left( 1 - 2 \sin^2 x \right) \left(1 - 2 \sin^2 x \right)

    = \left( 1 - 2 \sin^2 x \right)^2

    = \cos^2 2x
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