# How do you prove (1-\cos^2 x)(1+\cot^2 x) = 1?

Oct 24, 2014

By the trig identities

${\cos}^{2} x + {\sin}^{2} x = 1 R i g h t a r r o w {\sin}^{2} x = 1 - {\cos}^{2} x$

and

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

we have

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

I hope that this was helpful.