1. **Problem statement:** Simplify the expression $$\frac{\cos^2\left(\frac{\pi}{2} - x\right)}{\cos x}$$. 2. **Recall the co-function identity:** $$\cos\left(\frac{\pi}{2} - x\right) = \sin x$$. 3. Substitute the identity into the expression: $$\frac{\cos^2\left(\frac{\pi}{2} - x\right)}{\cos x} = \frac{\sin^2 x}{\cos x}$$. 4. This can be rewritten as: $$\frac{\sin^2 x}{\cos x} = \sin x \cdot \frac{\sin x}{\cos x} = \sin x \tan x$$. 5. **Answer:** The simplified form is $$\sin x \tan x$$. --- 6. **Problem statement:** Simplify $$\sin \phi (\csc \phi - \sin \phi)$$. 7. Recall that $$\csc \phi = \frac{1}{\sin \phi}$$. 8. Substitute: $$\sin \phi \left(\frac{1}{\sin \phi} - \sin \phi\right) = \sin \phi \left(\frac{1 - \sin^2 \phi}{\sin \phi}\right)$$. 9. Cancel $$\sin \phi$$: $$\sin \phi \cdot \frac{1 - \sin^2 \phi}{\sin \phi} = \cancel{\sin \phi} \cdot \frac{1 - \sin^2 \phi}{\cancel{\sin \phi}} = 1 - \sin^2 \phi$$. 10. Use Pythagorean identity $$1 - \sin^2 \phi = \cos^2 \phi$$. 11. **Answer:** The simplified form is $$\cos^2 \phi$$. --- 12. **Problem statement:** Simplify $$\frac{\sec^2 x - 1}{\sec x - 1}$$. 13. Recall the identity $$\sec^2 x - 1 = \tan^2 x$$. 14. Substitute: $$\frac{\tan^2 x}{\sec x - 1}$$. 15. Express $$\sec x = \frac{1}{\cos x}$$: $$\frac{\tan^2 x}{\frac{1}{\cos x} - 1} = \frac{\tan^2 x}{\frac{1 - \cos x}{\cos x}} = \tan^2 x \cdot \frac{\cos x}{1 - \cos x}$$. 16. Recall $$\tan x = \frac{\sin x}{\cos x}$$, so $$\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$$. 17. Substitute: $$\frac{\sin^2 x}{\cos^2 x} \cdot \frac{\cos x}{1 - \cos x} = \frac{\sin^2 x}{\cos x (1 - \cos x)}$$. 18. Factor numerator and denominator: $$\frac{\sin^2 x}{\cos x (1 - \cos x)}$$. 19. Use identity $$\sin^2 x = (1 - \cos x)(1 + \cos x)$$: $$\frac{(1 - \cos x)(1 + \cos x)}{\cos x (1 - \cos x)}$$. 20. Cancel $$1 - \cos x$$: $$\frac{\cancel{(1 - \cos x)} (1 + \cos x)}{\cos x \cancel{(1 - \cos x)}} = \frac{1 + \cos x}{\cos x} = 1 + \frac{1}{\cos x} = 1 + \sec x$$. 21. **Answer:** The simplified form is $$1 + \sec x$$. --- 22. **Problem statement:** Simplify $$1 - 2 \cos^2 x + \cos^4 x$$. 23. Recognize this as a perfect square: $$1 - 2 \cos^2 x + \cos^4 x = (1 - \cos^2 x)^2$$. 24. Use Pythagorean identity $$1 - \cos^2 x = \sin^2 x$$: $$ (\sin^2 x)^2 = \sin^4 x$$. 25. **Answer:** The simplified form is $$\sin^4 x$$. --- 26. **Problem statement:** Simplify $$\sin x \sec x$$. 27. Recall $$\sec x = \frac{1}{\cos x}$$. 28. Substitute: $$\sin x \cdot \frac{1}{\cos x} = \frac{\sin x}{\cos x} = \tan x$$. 29. **Answer:** The simplified form is $$\tan x$$. --- 30. **Problem statement:** Simplify $$\cos^2 x (\sec^2 x - 1)$$. 31. Recall $$\sec^2 x - 1 = \tan^2 x$$. 32. Substitute: $$\cos^2 x \tan^2 x$$. 33. Recall $$\tan x = \frac{\sin x}{\cos x}$$, so $$\tan^2 x = \frac{\sin^2 x}{\cos^2 x}$$. 34. Substitute: $$\cos^2 x \cdot \frac{\sin^2 x}{\cos^2 x} = \sin^2 x$$. 35. **Answer:** The simplified form is $$\sin^2 x$$. --- 36. **Problem statement:** Simplify $$\sec^4 x - \tan^4 x$$. 37. Recognize difference of squares: $$\sec^4 x - \tan^4 x = (\sec^2 x)^2 - (\tan^2 x)^2 = (\sec^2 x - \tan^2 x)(\sec^2 x + \tan^2 x)$$. 38. Recall identity $$\sec^2 x - \tan^2 x = 1$$. 39. Substitute: $$1 \cdot (\sec^2 x + \tan^2 x) = \sec^2 x + \tan^2 x$$. 40. **Answer:** The simplified form is $$\sec^2 x + \tan^2 x$$.