precalc

use power reducing identities to prove the identity

sin^4x=1/8(3-4cos2x+cos4x)

cos^3x=(1/2cosx) (1+cos2x)

thanks :)

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  1. cos 2x = 2cos^2 x - 1
    so, 1/2 cos x (1+2cos^2 x - 1) = cos^3 x

    cos 4x = 1 - 2sin^2 2x
    = 1 - 8sin^2 x cos^2 x
    = 1 - 8sin^2 x (1 - sin^2 x)
    = 1 - 8sin^2 x + 8 sin^4 x

    cos 2x = 1 - 2sin^2 x
    4cos 2x = 4 - 8sin^2 x

    1/8(3-4cos2x+cos4x)
    = 1/8(3 - 4 + 8sin^2 x + 1 - 8sin^2 x + 8 sin^4 x)
    = 1/8(8sin^4 x)
    = sin^4 x

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