Prove each identity using an algebraic approach. State restrictions. 4) $\frac{\csc x}{\sec ^{2} x+\csc ^{2} x}=\frac{\sin x}{\sec ^{2} x}$

Step 1 ：Rewrite the given expression in terms of sine and cosine functions: \(\frac{1/\sin x}{(1/\cos^2 x) + (1/\sin^2 x)}\)

Step 2 ：Simplify the expression: \(\frac{\sin x}{\cos^2 x}\)

Step 3 ：The simplified expression is equal to the right-hand side of the equation, so the identity is true: \(\boxed{\frac{\csc x}{\sec^2 x + \csc^2 x} = \frac{\sin x}{\sec^2 x}}\)