Pull out the common factor of [tex]\cos^{-1/2}x\sin x[/tex]:
[tex]\cos^{-1/2}x\sin x-\cos^{3/2}x\sin x=\cos^{-1/2}x\sin x(1-\cos^2x)[/tex]
Recall that [tex]1-\cos^2x=\sin^2x[/tex]:
[tex]\cos^{-1/2}x\sin x-\cos^{3/2}x\sin x=\dfrac{\sin^3x}{\sqrt{\cos x}}[/tex]
Rationalize the denominator:
[tex]\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}[/tex]
