To simplify the given trigonometric expression:

$\tan x \csc x$

Step 1: Express in Terms of Sine and Cosine

We use the definitions:

$\tan x = \frac{\sin x}{\cos x}, \quad \csc x = \frac{1}{\sin x}$

Substituting these:

$\tan x \csc x = \left(\frac{\sin x}{\cos x} \right) \times \left(\frac{1}{\sin x} \right)$

Step 2: Simplify

Cancel $\sin x$ in the numerator and denominator:

$\frac{\sin x}{\cos x} \times \frac{1}{\sin x} = \frac{1}{\cos x}$

Step 3: Express as a Single Trigonometric Function

Since $\frac{1}{\cos x} = \sec x$, we get:

$\tan x \csc x = \sec x$

Thus, the simplified expression is:

$\boxed{\sec x}$

Would you like more details or another example?

Here are five related questions to explore further:

1. How do you simplify $\sin x \tan x$ into a single trigonometric function?
2. What is the relationship between secant and cosine?
3. How do you prove $\sec^2 x - \tan^2 x = 1$?
4. Can you express $\cot x \csc x$ as a single function?
5. How do you derive $\tan x$ in terms of sine and cosine?

Tip: When simplifying trigonometric expressions, converting to sine and cosine often helps.