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 $1-\cos^2x=\sin^2x$:

$\cos^{-1/2}x\sin x-\cos^{3/2}x\sin x=\dfrac{\sin^3x}{\sqrt{\cos x}}$

Rationalize the denominator:

$\cos^{-1/2}x\sin x-\cos^{3/2}x\sin x=\dfrac{\sin^3x\sqrt{\cos x}}{(\sqrt{\cos x})^2}=\dfrac{\sin^3x\sqrt{\cos x}}{\cos x}$