Pre-Cal

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verify the identity

sin x(1 - 2 cos^2 x + cos^4 x) = sin^5 x

1-2cos^2 x looks like 2cos^2x-1 its just backwards. I am not sure where to start.

  • Pre-Cal -

    good observation
    recall that cos 2A = 2cos^2 A - 1 = 1 - 2sin^2 A

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


    LS
    =sin x(1 - 2 cos^2 x + cos^4 x)
    = sinx(2sin^2 x - 1 + (cos^2x)(cos^2x)
    = sinx( 2 sin^2x - 1 + (1-sin^2x)(1 -sin^2x))
    = sinx(2sin^2x - 1 + 1 - 2sin^2x + sin^4x)
    = sinx(sin^4x)
    = sin^5 x
    = RS

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