Question

How do you verify the identity of: `cos^3x sin^2x=(sin^2x-sin^4x)cosx`?

Answer 1

From basic definitions and the Pythagorean Theorem
`cos^2(x)+sin^2(x)=1`
or
`cos^2(x) = 1-sin^2(x)`

First consider
`(sin^2(x)-sin^4(x)`

`=(sin^2(x))*(1-sin^2(x))`

`=underbrace(sin^2(x)cos^2(x))_(used below)`

So
`cos^3(x)sin^2(x)`

`=(cos(x)) * [underbrace((cos^2(x)sin^2(x)))_(as above)]`

`= (cos(x))*(sin^2(x)-sin^4(x))`

`= (sin^2(x)-sin^4(x))cos(x)`