Question

Question #32396

Answer 1

The equation as given is invalid.
See below for a proof that assumes the exponents were omitted and the equation should be:
`sin(x)-sin(x)cos^2(x)=sin^3(x)`

Explanation:

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

factoring:
`color(white)("XXX")=sin(x)*(1-cos^2(x))`

Since `sin^2(x)+cos^2(x)=1`
`color(white)("XXX")`we know this as a standard identity
`color(white)("XXX")`or we could develop it easily from Pythagorean theorem and definition of `sin` and `cos`

`sin(x)*(color(red)(1-cos^2(x)))`
`color(white)("XXX")=sin(x)*color(red)(sin^2(x))`

`color(white)("XXX")=sin^3(x)`