# Prove? cotx/(cscx-1)=(cscx+1)/cotx

Remember that $1 + {\cot}^{2} x = {\csc}^{2} x$.

#### Explanation:

$\cot \frac{x}{\csc x - 1} = \frac{\csc x + 1}{\cot} x$

Remember that $1 + {\cot}^{2} x = {\csc}^{2} x$. This becomes useful if we multiply the terms with $\csc x$ to get them squared:

$\cot \frac{x}{\csc x - 1} \left(\frac{\csc x + 1}{\csc x + 1}\right) = \frac{\csc x + 1}{\cot} x$

$\frac{\cot x \csc x + \cot x}{{\csc}^{2} x - 1} = \frac{\csc x + 1}{\cot} x$

We can now use ${\csc}^{2} x - 1 = {\cot}^{2} x$

$\frac{\cot x \csc x + \cot x}{{\cot}^{2} x} = \frac{\csc x + 1}{\cot} x$

$\frac{\csc x + 1}{\cot x} = \frac{\csc x + 1}{\cot} x$