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

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Answer

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

Steps

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: $\boxed{\frac{\csc x}{\sec^2 x + \csc^2 x} = \frac{\sin x}{\sec^2 x}}$

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

Answer

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