1. **State the problem:** Find the derivative of the function $f(x) = \frac{1}{\tan^2 x}$. 2. **Rewrite the function:** Note that $\frac{1}{\tan^2 x} = \cot^2 x$. So, $f(x) = \cot^2 x$. 3. **Recall the derivative formula:** The derivative of $\cot x$ is $-\csc^2 x$. Using the chain rule, the derivative of $\cot^2 x$ is $2 \cot x \cdot \frac{d}{dx}(\cot x) = 2 \cot x (-\csc^2 x)$. 4. **Calculate the derivative:** $$f'(x) = 2 \cot x (-\csc^2 x) = -2 \cot x \csc^2 x.$$ 5. **Final answer:** $$\boxed{f'(x) = -2 \cot x \csc^2 x}.$$