Question

Can anyone prove this identity? Thank you.

Answer 1

Step-by-step explanation:

sin 3θ = sin(2θ + θ)

Angle sum formula:

= sin(2θ) cos θ + cos(2θ) sin θ

Double angle formulas:

= (2 sin θ cos θ) cos θ + (1 − 2 sin² θ) sin θ

Distribute:

= 2 sin θ cos² θ + sin θ − 2 sin³ θ

Pythagorean identity:

= 2 sin θ (1 − sin² θ) + sin θ − 2 sin³ θ

Distribute again:

= 2 sin θ − 2 sin³ θ + sin θ − 2 sin³ θ

Combine like terms:

= 3 sin θ − 4 sin³ θ