The problem asks you to use the reciprocal identities to solve for $\cot \theta$ when $\tan \theta = a$ (with $a \neq 0$).

We know the reciprocal identity between tangent and cotangent is: $\cot \theta = \frac{1}{\tan \theta}$

Given that $\tan \theta = a$, we substitute this into the equation: $\cot \theta = \frac{1}{a}$

Thus, the solution is: $\cot \theta = \frac{1}{a}$

Would you like further details or have any questions? Here are five related questions you might find helpful:

1. What are the reciprocal identities for sine, cosine, and tangent?
2. How do you derive the reciprocal identity for cotangent from the definition of tangent?
3. Can you express $\cot \theta$ in terms of sine and cosine?
4. How do you solve for $\theta$ if both $\tan \theta$ and $\cot \theta$ are given?
5. What happens to $\cot \theta$ when $\tan \theta$ approaches zero?

Tip: Always remember that reciprocal identities are useful for simplifying trigonometric expressions.