Let's solve the given problem step by step.

We are tasked with simplifying:

$\frac{\cot x \sec x}{\csc x}.$

Step 1: Express trigonometric functions in terms of $\sin x$ and $\cos x$

• $\cot x = \frac{\cos x}{\sin x}$,
• $\sec x = \frac{1}{\cos x}$,
• $\csc x = \frac{1}{\sin x}$.

Substituting these into the expression:

$\frac{\cot x \sec x}{\csc x} = \frac{\left(\frac{\cos x}{\sin x}\right) \left(\frac{1}{\cos x}\right)}{\frac{1}{\sin x}}.$

Step 2: Simplify the numerator

In the numerator, $\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x}$: $\frac{\cos x}{\sin x} \cdot \frac{1}{\cos x} = \frac{1}{\sin x}.$

So the expression becomes: $\frac{\frac{1}{\sin x}}{\frac{1}{\sin x}}.$

Step 3: Simplify the fraction

Dividing $\frac{1}{\sin x}$ by $\frac{1}{\sin x}$: $\frac{\frac{1}{\sin x}}{\frac{1}{\sin x}} = 1.$

Final Answer:

$\boxed{1}.$

Let me know if you have further questions or want additional details!

Related Questions:

1. What is the general approach to simplify trigonometric expressions?
2. How do you derive reciprocal trigonometric identities?
3. What are the applications of $\sec x, \csc x$ in simplifying problems?
4. Why does $\frac{1}{x}$ divided by $\frac{1}{x}$ simplify to 1?
5. What are the domain restrictions for $\cot x$ and $\csc x$?

Tip:

Always start simplifying trigonometric expressions by rewriting everything in terms of $\sin x$ and $\cos x$ for consistency.