Answer
$$\cot\theta=\sqrt {\frac{1}{\sin^{2}\theta}-1}$$
Work Step by Step
Start with the Pythagorean Identity
$$\csc^{2}\theta=1+\cot^{2}\theta.$$
Rearrange terms and solve for $\cot\theta$:
$$\cot^{2}\theta=\csc^{2}\theta-1$$
$$\cot\theta=\sqrt {\csc^{2}\theta-1}.$$
Using the Reciprocal Identity
$$\sin\theta=\frac{1}{\csc\theta},$$
substitute to obtain
$$\cot\theta=\sqrt {\frac{1}{\sin^{2}\theta}-1}.$$
