Question

Prove the identity and give its domain.

4sinxcosx+sin4x=8sinxcos^3x

Answer 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:
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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.