We are tasked with simplifying the following trigonometric expression:

$\cos(-x) + \frac{\sin(-x)}{\cot(-x)}$

Let us proceed step by step:

Step 1: Apply trigonometric identities for negative angles

1. Cosine: $\cos(-x) = \cos(x)$ because cosine is an even function.
2. Sine: $\sin(-x) = -\sin(x)$ because sine is an odd function.
3. Cotangent: $\cot(-x) = -\cot(x)$ because cotangent is the ratio of cosine and sine ($\cot(x) = \frac{\cos(x)}{\sin(x)}$).

Substitute these into the expression:

$\cos(-x) + \frac{\sin(-x)}{\cot(-x)} = \cos(x) + \frac{-\sin(x)}{-\cot(x)}.$

Step 2: Simplify the fraction

Notice that the negatives in the numerator and denominator cancel out:

$\frac{-\sin(x)}{-\cot(x)} = \frac{\sin(x)}{\cot(x)}.$

Now the expression becomes:

$\cos(x) + \frac{\sin(x)}{\cot(x)}.$

Step 3: Replace $\cot(x)$ with its definition

The cotangent function is defined as:

$\cot(x) = \frac{\cos(x)}{\sin(x)}.$

Substitute this into the fraction:

$\frac{\sin(x)}{\cot(x)} = \frac{\sin(x)}{\frac{\cos(x)}{\sin(x)}}.$

Simplify the complex fraction by multiplying by the reciprocal:

$\frac{\sin(x)}{\frac{\cos(x)}{\sin(x)}} = \sin(x) \cdot \frac{\sin(x)}{\cos(x)} = \frac{\sin^2(x)}{\cos(x)}.$

Step 4: Combine terms over a common denominator

Now the expression is:

$\cos(x) + \frac{\sin^2(x)}{\cos(x)}.$

To add these terms, write $\cos(x)$ with a common denominator:

$\cos(x) = \frac{\cos^2(x)}{\cos(x)}.$

Thus:

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

Combine the numerators:

$\frac{\cos^2(x) + \sin^2(x)}{\cos(x)}.$

Step 5: Use the Pythagorean identity

The Pythagorean identity states:

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

Substitute this into the expression:

$\frac{\cos^2(x) + \sin^2(x)}{\cos(x)} = \frac{1}{\cos(x)}.$

Step 6: Simplify to a single trigonometric function

The reciprocal of cosine is secant ($\sec(x)$):

$\frac{1}{\cos(x)} = \sec(x).$

Final Answer:

The simplified expression is:

$\boxed{\sec(x)}.$

Do you want me to explain any step further or provide more examples? Let me know! Here are five related questions for further practice:

1. Simplify $\cos(-x) + \sin(-x)$ using trigonometric identities.
2. Simplify $\tan(-x) + \cot(-x)$.
3. Prove that $\sin^2(x) + \cos^2(x) = 1$ using the unit circle definition.
4. Simplify $\frac{\sin^2(x)}{1 - \cos(x)}$.
5. Simplify $\frac{1 - \cos(2x)}{\sin(2x)}$.

Tip: Always check for symmetry and parity properties (even or odd functions) when simplifying trigonometric expressions.