To verify the identity $\frac{\cot x}{\sec x} = \csc x - \sin x$, let's simplify the left-hand side (LHS) and right-hand side (RHS) separately.

Step 1: Express LHS in terms of basic trigonometric functions

$\frac{\cot x}{\sec x} = \frac{\frac{\cos x}{\sin x}}{\frac{1}{\cos x}} = \cos x \cdot \frac{\cos x}{\sin x} = \frac{\cos^2 x}{\sin x}$

Step 2: Simplify RHS

$\csc x - \sin x = \frac{1}{\sin x} - \sin x$ Convert the terms to a common denominator: $\csc x - \sin x = \frac{1 - \sin^2 x}{\sin x}$ Use the Pythagorean identity $1 - \sin^2 x = \cos^2 x$: $\csc x - \sin x = \frac{\cos^2 x}{\sin x}$

Since the LHS and RHS are equal, the identity is verified:

$\frac{\cot x}{\sec x} = \csc x - \sin x$

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Related Questions:

1. How do you prove trigonometric identities?
2. What are the fundamental trigonometric identities?
3. How do you simplify expressions involving trigonometric functions?
4. What is the Pythagorean identity in trigonometry?
5. How do you solve equations involving multiple trigonometric functions?

Tip: Always convert trigonometric functions to basic terms (sine and cosine) when proving identities.