Let's solve these expressions step-by-step.

Problem 1:

$\frac{\cos(-\alpha) - \sin(-\alpha)}{\tan(\alpha)}$

Solution:

1. Use trigonometric identities:

   • $\cos(-\alpha) = \cos(\alpha)$ (cosine is an even function).
   • $\sin(-\alpha) = -\sin(\alpha)$ (sine is an odd function).
   • $\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$.

2. Substitute these identities:

   = \frac{\cos(\alpha) + \sin(\alpha)}{\frac{\sin(\alpha)}{\cos(\alpha)}}$$

3. Simplify by multiplying both numerator and denominator by $\cos(\alpha)$:

   = \cos^2(\alpha) + \cos(\alpha)\sin(\alpha)$$

However, the problem asks to simplify using a single trig function. Further simplification may involve recognizing that this expression can be difficult to reduce to a single function.

Let's verify the specific context or ask for clarification if there are specific guidelines on the simplification level.

Problem 2:

$\tan(-\alpha)\sin(-\alpha) + \cos(-\alpha)$

Solution:

1. Use trigonometric identities:

   • $\tan(-\alpha) = -\tan(\alpha)$.
   • $\sin(-\alpha) = -\sin(\alpha)$.
   • $\cos(-\alpha) = \cos(\alpha)$.

2. Substitute these identities: $(-\tan(\alpha)) \cdot (-\sin(\alpha)) + \cos(\alpha)$ Simplify by multiplying: $= \tan(\alpha) \sin(\alpha) + \cos(\alpha)$

3. Substitute $\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$:

   = \frac{\sin^2(\alpha)}{\cos(\alpha)} + \cos(\alpha)$$

4. Combine terms using a common denominator: $= \frac{\sin^2(\alpha) + \cos^2(\alpha)}{\cos(\alpha)}$ Using the Pythagorean identity $\sin^2(\alpha) + \cos^2(\alpha) = 1$: $= \frac{1}{\cos(\alpha)} = \sec(\alpha)$

So, the answer for Problem 2 is: $\sec(\alpha)$

Would you like further details on any of these steps or explanations?

Here are some related questions:

1. What are even and odd trigonometric functions, and why do they matter?
2. How do you apply Pythagorean identities to simplify trigonometric expressions?
3. What are the properties of tangent and secant functions?
4. Why is $\tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)}$ essential in simplifying expressions?
5. How can complex trigonometric expressions sometimes be simplified to a single trigonometric function?

Tip: Knowing the basic identities like Pythagorean, even-odd, and reciprocal identities can significantly simplify many trigonometric expressions.