The function you're working with is \( f(z) = z^2 \cot(z) \), and you are looking for the zeros of this function.

To find the zeros of \( f(z) = z^2 \cot(z) \), we must find values of \( z \) such that:

\[
z^2 \cot(z) = 0
\]

Since this is a product, the function will be zero if either factor is zero:

1. \( z^2 = 0 \)
2. \( \cot(z) = 0 \)

### 1. Solving \( z^2 = 0 \)
This equation gives us the solution:
\[
z = 0
\]

### 2. Solving \( \cot(z) = 0 \)
The cotangent function \( \cot(z) \) is zero whenever \( z = \frac{(2n+1)\pi}{2} \) for integers \( n \). These are the points where the sine function is 1 or -1 (since \( \cot(z) = \frac{\cos(z)}{\sin(z)} \)).

Thus, the solutions are:
\[
z = \frac{(2n+1)\pi}{2}, \quad n \in \mathbb{Z}
\]

### Conclusion
The zeros of \( f(z) = z^2 \cot(z) \) are:
\[
z = 0 \quad \text{and} \quad z = \frac{(2n+1)\pi}{2}, \quad n \in \mathbb{Z}
\]

Would you like further details on the solution?

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Here are 5 related questions:
1. What are the properties of the cotangent function?
2. How would the graph of \( f(z) = z^2 \cot(z) \) look near the zeros?
3. What happens to \( f(z) \) at the points where \( \cot(z) \) is undefined?
4. Can this problem be extended to complex numbers \( z \)?
5. How do the zeros of \( \cot(z) \) relate to the zeros of \( \sin(z) \)?

**Tip:** Always check the domain of trigonometric functions, as they can introduce undefined points in the function.

Explore how to find the zeros of the function z^2 cot(z) with a step-by-step solution. This problem involves key concepts from trigonometry, such as the cotangent function and the zeros of trigonometric functions. Suitable for college-level students.

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