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Math Help - Trigonometric expression

  1. #1
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    Trigonometric expression

    hi everyone.
    I need to consider the following expression:
    sin(x)^4 - cos(x)^4 + 2cos(x)^2 = 1

    I need to also use following identities to show by hand that this one above is correct: sin^2(x) + cos^2(x) = 1 and a^2 - b^2 = (a - b)(a + b)

    Thx for any help.
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  2. #2
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    Quote Originally Posted by Snowboarder View Post
    hi everyone.
    I need to consider the following expression:
    sin(x)^4 - cos(x)^4 + 2cos(x)^2 = 1

    I need to also use following identities to show by hand that this one above is correct: sin^2(x) + cos^2(x) = 1 and a^2 - b^2 = (a - b)(a + b)

    Thx for any help.
    Note that \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x).
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  3. #3
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    Also, you need to know that:
    \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}

    \cos{2\theta} = 2\cos^2{\theta} - 1

    Just manipulate and substitute.
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  4. #4
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    Quote Originally Posted by Chop Suey View Post
    Also, you need to know that:
    \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta}

    \cos{2\theta} = 2\cos^2{\theta} - 1

    Just manipulate and substitute.
    No you don't.

    \sin^4 x - \cos^4 x = (\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) = \sin^2 x - \cos^2 x.

    Then Left Hand Side = \sin^2 x - \cos^2 x + 2 \cos^2 x = \sin^2 x + \cos^2 x = 1 = Right Hand Side.
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