We are tasked with simplifying the expression:

$\coth(x) + \coth(x) \csc^2(x)$

Step 1: Factor $\coth(x)$

We can factor out $\coth(x)$ from both terms:

$\coth(x) + \coth(x) \csc^2(x) = \coth(x) \left(1 + \csc^2(x)\right)$

Step 2: Use a known identity

Recall that $\coth(x)$ is defined as:

$\coth(x) = \frac{\cosh(x)}{\sinh(x)}$

and $\csc(x)$ is related to sine:

$\csc(x) = \frac{1}{\sin(x)}$

However, since there is no direct identity involving both coth and $\csc^2(x)$ that simplifies further in this form, the expression is already factored and simplified.

Thus, the simplified form of the given expression is:

$\boxed{\coth(x) \left(1 + \csc^2(x)\right)}$

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

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Here are some related questions you might want to explore:

1. What is the derivative of $\coth(x)$?
2. How is $\csc(x)$ related to the other trigonometric functions?
3. What is the integral of $\coth(x)$?
4. How can hyperbolic functions be used in real-world applications?
5. Can you simplify $\csch(x) + \csch(x)\csc(x)$?

Tip: Hyperbolic functions like $\coth(x)$ are analogous to trigonometric functions and can be useful in many areas of mathematics, especially in calculus and differential equations.