# How would you prove or disprove cotx - cosx/cotx = cos^2x/(1 + sinx)?

Oct 30, 2016

Rewrite all terms in $\cot x$ as $\cos \frac{x}{\sin} x$.

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

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

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

Rewrite the right-hand side using the identity ${\sin}^{2} x + {\cos}^{2} x = 1$.

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

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

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

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

The identity is false, because no matter what you do with the left hand side, you will never be able to get on the right.

Hopefully this helps!