Prove that sin^5x=sinx-2cos^2xsinx+cos^4xsinx

Question

sin^5x=sinx-2cos^2xsinx+cos^4xsinx

Please prove

Answer 1

The above answer is somewhat confusing.

sin(x)-2sin(x)cos^2(x)+sin(x)cos^4(x)

= sin(x)[1 - 2cos^2(x) + cos^4(x)].

The above expression is sin(x) multiplied by a perfect square.

We factor that perfect square:

= sin(x)[1 - cos^2(x)]^2

= sin(x) [sin^2(x)]^2

= sin(x) sin^4(x) = sin^5(x).

Answer 2

Work on the right side of the equation

Factor out a sinx

sinx (1-2cos^2x+cos^4x)

Rearrange

Sinx (cos^4x-2cos^2x+1)

Factor

Sinx (cos^2x-1)(cos^2x-1)

Substitute the identity

Sinx (sin^2x) (sin^2x)

Sin^5x