Calculus

Prove the identity and give its domain.

4sinxcosx+sin4x=8sinxcos^3x

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  1. Use the identities
    sin 2y = 2 sin y cosy,
    cis 2y = cos^2 y - sin^2 y. and
    1 - sin^2 y = cos^2 y
    as follows:
    =================
    4sinxcosx + sin4x = 8sinxcos^3x
    2sin2x + sin4x = (8sinxcosx)cos^2x
    2sin2x + 2sin2x cos2x = 4 sin2x cos^2x
    Divide both sides by 2sin2x
    1 + cos 2x = 2 cos^2 x
    1 + cos^2x - sin^2x = 2 cos^2x
    Subtract cos^2x from both sides
    1 - sin^2 x = cos^2 x
    etc.
    You may need to prove separately that the identity is valid when sin2x = 0, since we divided by that term.

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