Let's analyze and match each expression with the correct identity.

39. $-\cot(x)\sin(x) = ?$

This expression simplifies using trigonometric identities: $-\cot(x)\sin(x) = -\frac{\cos(x)}{\sin(x)} \cdot \sin(x) = -\cos(x)$ Answer: $-\cos(x)$

40. $\sec^2(x) - 1 = ?$

This is a well-known Pythagorean identity: $\sec^2(x) - 1 = \tan^2(x)$ Answer: $\tan^2(x)$

41. $\frac{\csc(x)}{\sec(x)} = ?$

Using the definitions of cosecant and secant: $\frac{\csc(x)}{\sec(x)} = \frac{\frac{1}{\sin(x)}}{\frac{1}{\cos(x)}} = \frac{\cos(x)}{\sin(x)} = \cot(x)$ Answer: $\cot(x)$

42. $1 + \sin^2(x) = ?$

This is not a standard identity. However, $1 + \sin^2(x)$ could remain as is unless further context is given. No simplification exists without more details.

Answer: $1 + \sin^2(x)$ (no simplification)

43. $\cos^2(x) = ?$

There is no need to rewrite this, as it is already a standard expression. However, depending on context, it could be part of the Pythagorean identity: $\cos^2(x) + \sin^2(x) = 1$ But in isolation, it remains $\cos^2(x)$.

Answer: $\cos^2(x)$ (no change)

Do you need any further details or explanations on these steps?

Here are 5 related questions:

1. How do you prove the identity $\sec^2(x) - 1 = \tan^2(x)$?
2. What is the relationship between cotangent and cosine?
3. How do you derive $\frac{\csc(x)}{\sec(x)} = \cot(x)$?
4. Can $1 + \sin^2(x)$ be rewritten under certain conditions?
5. How is the identity $\cos^2(x) + \sin^2(x) = 1$ applied in trigonometry?

Tip: The Pythagorean identities are essential tools in simplifying trigonometric expressions.