Question

Simplify using trigonometric identities
2sinθ - sin2θ cosθ

Answer 1

Double angle identity for sine:

`\sin(2x) = 2 \sin(x) \cos(x)`

`\implies 2 \sin(\theta) - \sin(2\theta) \cos(\theta) = 2 \sin(\theta) - 2 \sin(\theta) \cos^2(\theta)`

Factorize the left side.

`2 \sin(\theta) - 2 \sin(\theta) \cos^2(\theta) = 2 \sin(\theta) \left(1 - \cos^2(\theta)\right)`

Pythagorean identity:

`\cos^2(x) + \sin^2(x) = 1`

`\implies 2 \sin(\theta) \left(1 - \cos^2(\theta)\right) = 2 \sin^3(\theta)`

so that

`\boxed{2 \sin(\theta) - \sin(2\theta) \cos(\theta) = 2 \sin^3(\theta)}`