Question

Cos^-1/2xsinx-cos^3/2xsinx=sin^3x√cosx/cosx​

Answer 1

Pull out the common factor of
`\cos^(-1/2)x\sin x`:

`\cos^(-1/2)x\sin x-\cos^(3/2)x\sin x=\cos^(-1/2)x\sin x(1-\cos^2x)`

Recall that
`1-\cos^2x=\sin^2x`:

`\cos^(-1/2)x\sin x-\cos^(3/2)x\sin x=(\sin^3x)/(√(\cos x))`

Rationalize the denominator:

`\cos^(-1/2)x\sin x-\cos^(3/2)x\sin x=(\sin^3x√(\cos x))/((√(\cos x))^2)=(\sin^3x√(\cos x))/(\cos x)`