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Chapter 5: Problem 28

Verify the identity. $$\frac{1}{\sin x}-\frac{1}{\csc x}=\csc x-\sin x$$

Short Answer

Expert verified
Yes, the trigonometric identity \(\frac{1}{\sin x} - \frac{1}{\csc x} = \csc x - \sin x\) provided in the exercise holds true.

Step by step solution

01

Simplify Left Side

Start with the left-hand side of the equation, \(\frac{1}{\sin x} - \frac{1}{\csc x}\). As mentioned, \(\csc x\) equals \(1 / \sin x\), so they are reciprocal. Hence, \(\frac{1}{\sin x} - \frac{1}{\csc x}\) simplifies to zero.
02

Simplify Right Side

Next, consider the right-hand side of the equation, which is \(\csc x - \sin x\). As \(\csc x = 1 / \sin x\), the right side can be rewritten as \(1 / \sin x - \sin x\). It can further simplify to \(0\), same as the left side.
03

Comparing both sides

Finally, after simplifying both sides, we have zero equals zero, which means the original equation indeed hold true, and the identity is verified.

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