The expression you're asking about is:

$\cot^2 \theta \left( 1 + \tan^2 \theta \right)$

To simplify this, let's use some trigonometric identities:

1. Pythagorean identity: $1 + \tan^2 \theta = \sec^2 \theta$

Substituting this into the given expression:

$\cot^2 \theta \cdot \sec^2 \theta$

Next, recall the identity for cotangent and secant in terms of sine and cosine:

$\cot \theta = \frac{\cos \theta}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}$

Thus:

$\cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta}, \quad \sec^2 \theta = \frac{1}{\cos^2 \theta}$

Substituting these into the expression:

$\frac{\cos^2 \theta}{\sin^2 \theta} \cdot \frac{1}{\cos^2 \theta}$

The $\cos^2 \theta$ terms cancel out, leaving:

$\frac{1}{\sin^2 \theta}$

Which is:

$\csc^2 \theta$

So, the simplified form of your expression is:

$\csc^2 \theta$

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

Here are five related questions you might find interesting:

1. How do the identities for cotangent and secant help in simplifying trigonometric expressions?
2. What are other ways to prove the identity $1 + \tan^2 \theta = \sec^2 \theta$?
3. Can this method be applied to other trigonometric identities?
4. How do you convert between sine, cosine, and other trigonometric functions?
5. What are some practical applications of trigonometric identities in physics or engineering?

Tip: When simplifying trigonometric expressions, always consider using the fundamental identities (like the Pythagorean identities) to reduce the complexity.