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Let's simplify the given expression:

\[
\cos^2{x} \left( 1 + \cot^2{x} \right)
\]

We can use trigonometric identities to simplify it.

### Step 1: Use the Pythagorean identity for \(\cot^2{x}\)
The identity is:

\[
1 + \cot^2{x} = \csc^2{x}
\]

This transforms the expression into:

\[
\cos^2{x} \cdot \csc^2{x}
\]

### Step 2: Express \(\csc{x}\) in terms of \(\sin{x}\)
We know that:

\[
\csc{x} = \frac{1}{\sin{x}}
\]

Therefore, \(\csc^2{x} = \frac{1}{\sin^2{x}}\).

Now the expression becomes:

\[
\cos^2{x} \cdot \frac{1}{\sin^2{x}}
\]

### Step 3: Use the identity \(\frac{\cos{x}}{\sin{x}} = \cot{x}\)

So:

\[
\frac{\cos^2{x}}{\sin^2{x}} = \cot^2{x}
\]

Thus, the simplified form of the expression is:

\[
\cot^2{x}
\]

---

Would you like more details, or do you have any further questions on this simplification?

### Relative questions:
1. What are the main trigonometric identities used in simplifying expressions?
2. How does the identity \(1 + \cot^2{x} = \csc^2{x}\) derive from the Pythagorean identity?
3. Can you use a similar method to simplify \(\sin^2{x}(1 + \tan^2{x})\)?
4. How do you prove that \(\frac{\cos^2{x}}{\sin^2{x}} = \cot^2{x}\)?
5. What are the practical applications of simplifying trigonometric expressions?

### Tip:
Always try to recall the fundamental trigonometric identities like the Pythagorean identities—they often simplify complex-looking problems efficiently.

Simplify cos²(x)(1 + cot²(x)) using algebra and fundamental trigonometric identities.

Learn how to simplify the expression cos²(x)(1 + cot²(x)) by using fundamental trigonometric identities like Pythagorean identities. This step-by-step guide is suitable for high school students in Grades 10-12.

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