Let's simplify the expression $\frac{\tan(x) + \cot(x)}{\csc(x)}$ in terms of $\cos(x)$.

1. Step 1: Rewrite the trigonometric functions in terms of sine and cosine. $\tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}, \quad \csc(x) = \frac{1}{\sin(x)}$

2. Step 2: Substitute these into the original expression. $\frac{\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}} = \frac{\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}}{\frac{1}{\sin(x)}}$

3. Step 3: Simplify the numerator. Combine the two terms in the numerator: $\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)}$ Since $\sin^2(x) + \cos^2(x) = 1$ (Pythagorean identity), the numerator becomes: $\frac{1}{\sin(x)\cos(x)}$

4. Step 4: Divide by $\frac{1}{\sin(x)}$. Now divide by $\csc(x) = \frac{1}{\sin(x)}$: $\frac{\frac{1}{\sin(x)\cos(x)}}{\frac{1}{\sin(x)}} = \frac{1}{\sin(x)\cos(x)} \times \sin(x) = \frac{1}{\cos(x)}$

5. Final Answer: $\frac{1}{\cos(x)}$ This simplifies to $\sec(x)$.

Thus, the simplified form of $\frac{\tan(x) + \cot(x)}{\csc(x)}$ in terms of $\cos(x)$ is $\sec(x)$.

Would you like more details on any step or clarification?

Here are five related questions you can explore:

1. How is the secant function related to cosine?
2. What are other Pythagorean identities and how are they used?
3. How can you rewrite other trigonometric functions in terms of cosine?
4. What is the domain of the $\sec(x)$ function?
5. How does simplifying trigonometric expressions help in calculus?

Tip: When simplifying trigonometric expressions, always try converting everything to sine and cosine—it makes the process more straightforward.