# How do you verify the identity sin^(1/2)xcosx-sin^(5/2)xcosx=cos^3xsqrtsinx?

Jan 6, 2017

$L H S = {\sin}^{\frac{1}{2}} x \cos x - {\sin}^{\frac{5}{2}} x \cos x$

$= {\sin}^{\frac{1}{2}} x \cos x - {\sin}^{2} x {\sin}^{\frac{1}{2}} x \cos x$

$= {\sin}^{\frac{1}{2}} x \cos x \left(1 - {\sin}^{2} x\right)$

$= {\sin}^{\frac{1}{2}} x \cos x \left({\cos}^{2} x\right)$

$= {\cos}^{3} x \sqrt{\sin} x = R H S$

Verified