Verify that the equation is an identity.
\[
\frac{1+\cot ^{2} \beta}{\csc \beta}=\csc \beta
\]
To verify the identity, start with the more complicated side and transform it to look like the other side. Choose the correct transformations and transform the expression at each step.
\[
\begin{aligned}
\frac{1+\cot ^{2} \beta}{\csc \beta} & =\frac{\square}{\csc \beta} \\
& =\csc \beta
\end{aligned}
\]
Answer
Final Answer: \(\boxed{\text{True}}\)
Step 1 ：Verify that the equation is an identity: \(\frac{1+\cot ^{2} \beta}{\csc \beta}=\csc \beta\)
Step 2 ：Start with the more complicated side of the equation, which is the left side: \(\frac{1+\cot ^{2} \beta}{\csc \beta}\)
Step 3 ：Use the Pythagorean identity to replace \(1 + \cot^2\beta\) with \(\csc^2\beta\), resulting in \(\frac{\csc^2\beta}{\csc\beta}\)
Step 4 ：Simplify \(\frac{\csc^2\beta}{\csc\beta}\) to \(\csc\beta\)
Step 5 ：Since the transformed left side of the equation now matches the right side, the equation is an identity
Step 6 ：Final Answer: \(\boxed{\text{True}}\)
