1/cosx - 1/secx = sinxtanx (prove the identity with the rule)

Solution:
1. Write the identity to be proven:
- \frac{1}{\cos x} - \frac{1}{\sec x} = \sin x \tan x

2. Express everything in sine and cosine:
- \sec x = \frac{1}{\cos x}
- Therefore, \frac{1}{\sec x} = \cos x

3. Substitute into the left side of the identity:
- \frac{1}{\cos x} - \cos x
- = \frac{1}{\cos x} - \frac{\cos^2 x}{\cos x}
- = \frac{1 - \cos^2 x}{\cos x}

4. Simplify the expression using the Pythagorean identity:
- 1 - \cos^2 x = \sin^2 x
- Substitute: \frac{\sin^2 x}{\cos x}

5. Rewrite the right-hand side using the definition of tangent:
- \sin x \tan x = \sin x \cdot \frac{\sin x}{\cos x}
- = \frac{\sin^2 x}{\cos x}

6. Compare both sides:
- \frac{\sin^2 x}{\cos x} = \frac{\sin^2 x}{\cos x}

7. The identity is proven.