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Math Help - Trig Identity

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    Trig Identity

    Where do I even start?

    4(sin^2x)(cos^2x) + cos^2(2x) = 1
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    Quote Originally Posted by millerst View Post
    Where do I even start?

    4(sin^2x)(cos^2x) + cos^2(2x) = 1
    Hint: \sin 2x = 2 \sin x \cos x ....what happens if we square this equation?
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    Quote Originally Posted by Jhevon View Post
    Hint: \sin 2x = 2 \sin x \cos x ....what happens if we square this equation?
    Then I can use an identity, but would it make more sense to factor out the cosx or something... I'm so lost...

    or cos2X = cos^2x - sin^2x? so it becomes cos^4x - sin^4x?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by millerst View Post
    Then I can use an identity, but would it make more sense to factor out the cosx or something... I'm so lost...

    or cos2X = cos^2x - sin^2x? so it becomes cos^4x - sin^4x?
    please reread my post (note that i did not mention cos(2x)) and answer the question i asked


    by the way, you squared your identity incorrectly: (\cos^2 x - \sin^2 x)^2 {\color{red}\ne} \cos^4 x - \sin^4 x, since we (should) know that (a + b)^2 = a^2 + 2ab + b^2
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    Quote Originally Posted by Jhevon View Post
    Hint: \sin 2x = 2 \sin x \cos x ....what happens if we square this equation?

    \sin 2x = 2 \sin x \cos x so squared it becomes:

    4\sin^2x\cos^2x?

    Oh I see...


    So \sin 2x = 2 \sin x \cos x + cos^2/2x = 1
    \sin 2x = 2 \sin x \cos x + (cos2x)^2 = 1
    So 2 \sin x \cos x + (cos^2 - sin^2)^2 = 1
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by millerst View Post
    \sin 2x = 2 \sin x \cos x so squared it becomes:

    4\sin^2x\cos^2x?

    Oh I see...


    So \sin 2x = 2 \sin x \cos x + cos^2/2x = 1
    \sin 2x = 2 \sin x \cos x + (cos2x)^2 = 1
    So 2 \sin x \cos x + (cos^2 - sin^2)^2 = 1
    huh? you're thinking way too hard, Miller

    \begin{matrix} LHS & = & 4 \sin^2 x \cos^2 x + \cos^2 2x \\ & = & \sin^2 2x + \cos^2 2x \\ & = & 1 \\ & = & RHS \end{matrix}

    Thus, the identity is proved
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    How does

    sin^2(2x) + cos^2(2x) = 1
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  8. #8
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    Quote Originally Posted by millerst View Post
    How does

    sin^2(2x) + cos^2(2x) = 1
    The Pythagorean identity. Probably the most basic in trig.

    Let u = 2x for easier visualisation
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