Answer
Work on the left side of the identity using $\csc{\theta}=\frac{1}{\sin\theta}$ and $\cot\theta=\frac{\cos\theta}{\sin\theta}$.
Refer to the step-by-step part below for the complete proof.
Work Step by Step
We have to show that:
$(\csc\theta+1)(\csc\theta-1)=\cot^2\theta$
By evaluating the left side we get:
$=\csc^2\theta+\csc\theta-\csc\theta-1\\
=\csc^2\theta-1$
Since $\csc\theta=\frac{1}{\sin\theta}$ and $1-\sin^2\theta=\cos^2\theta$, the expression above simplifies to:
$=\dfrac{1}{\sin^2\theta}-\dfrac{\sin^2\theta}{\sin^2\theta}$
$=\dfrac{1-\sin^2\theta}{\sin^2\theta}$
$=\dfrac{\cos^2\theta}{\sin^2\theta}$
$=\cot^2\theta$
Thus, with LHS=RHS, the identity's proof is complete.
