Question

Identity transformations of Trigonometric Expressions.
prove the following identities.
$\frac{tan(x-\frac{\pi }{2})cos(\frac{3}{2}\pi +x)-si{n}^3(\frac{7}{2}\pi -x)}{cos(x-\frac{\pi }{2})tan(\frac{3}{2}\pi +x)}=si{n}^{2}x.$

Answer 1

$\frac{\tan (x-\frac{\pi }{2}).\cos (\frac{3\pi }{2}+x)-{\sin }^3(\frac{7\pi }{2}-x)}{\cos (x-\frac{\pi }{2}).\tan (\frac{3\pi }{2}+x)}$

$=\frac{(-\cot x).sinx+co{s}^{3}x}{\sin x.(-\cot x)}$

$=\frac{-\cos x+{\cos }^{3}x}{-\cos x}$

$=(1-co{s}^{2}x)$

$=si{n}^{2}x$