Question

Prove the following identity. sin 3θ = 3 sin θ − 4 sin3 θ

Answer 1

I'll work in x because I'm on my phone.

Proof: LHS = sin3x = sin(2x + x)
= sin2x cosx + cos2x sinx
= 2sinxcos^2x + cos^2x sinx - sin^3x
= 2sinx (1 - sin^2x) + sinx (1 - sin^2x) - sin^3x
= 2sinx - 2sin^3x + sinx - sin^3x - sin^3x
= 3sinx - 4sin^3x = RHS