Answer
$$\sin^2\beta(1+\cot^2\beta)=1$$
The trigonometric expression has been proved to be an identity. We simplify the left side first.
Work Step by Step
$$\sin^2\beta(1+\cot^2\beta)=1$$
We simplify the left side, since it is obviously more complex.
$$A=\sin^2\beta(1+\cot^2\beta)$$
We can change $1+\cot^2\beta$ according to the following Pythagorean Identity:
$$1+\cot^2\beta=\csc^2\beta$$
Also from a Reciprocal Identity, $$\csc\beta=\frac{1}{\sin\beta}$$
That means $$\csc^2\beta=\frac{1}{\sin^2\beta}$$
Therefore, $$1+\cot^2\beta=\frac{1}{\sin^2\beta}$$
We now apply it into $A$:
$$A=\sin^2\beta\times\frac{1}{\sin^2\beta}$$
$$A=1$$
The left side and right side are equal. The trigonometric expression is an identity.