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If sin x + cos x = a then sin⁶ x + cos⁶ x =

Mathematics

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Step-by-step solution

2 steps

  1. Square the given equation.

    We are given sin⁡x+cos⁡x=a\sin x + \cos x = a.
    Squaring both sides:

    (sin⁡x+cos⁡x)2=a2(\sin x + \cos x)^2 = a^2 sin⁡2x+cos⁡2x+2sin⁡xcos⁡x=a2\sin^2 x + \cos^2 x + 2 \sin x \cos x = a^2

    Using the identity sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1:

    1+2sin⁡xcos⁡x=a21 + 2 \sin x \cos x = a^2 2sin⁡xcos⁡x=a2−12 \sin x \cos x = a^2 - 1 sin⁡xcos⁡x=a2−12\sin x \cos x = \frac{a^2 - 1}{2}

  2. Express $\sin^6 x + \cos^6 x

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