We are tasked with proving that:

\[
\cot A = \frac{1}{\tan A}
\]

### Proof:

1. **Definition of cotangent and tangent**:
   The cotangent and tangent functions are defined as follows in terms of sine and cosine:

   \[
   \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A}
   \]

2. **Reciprocal relationship**:
   Notice that the cotangent function is the reciprocal of the tangent function. That is:

   \[
   \cot A = \frac{\cos A}{\sin A}
   \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A}
   \]

3. **Reciprocal of tangent**:
   Now, take the reciprocal of the tangent function:

   \[
   \frac{1}{\tan A} = \frac{1}{\frac{\sin A}{\cos A}} = \frac{\cos A}{\sin A}
   \]

4. **Equality**:
   Since \(\cot A = \frac{\cos A}{\sin A}\), we can conclude that:

   \[
   \frac{1}{\tan A} = \cot A
   \]

Thus, we have proven that:

\[
\cot A = \frac{1}{\tan A}
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the relationship between \(\cot A\) and \(\tan A\) on the unit circle?
2. How do you express other trigonometric functions in terms of cotangent?
3. How does the reciprocal identity apply to secant and cosine?
4. Can you derive \(\cot A\) from the Pythagorean identity?
5. What is the derivative of \(\cot A\) in calculus?

**Tip**: Trigonometric identities are key tools in simplifying expressions and solving equations. Keep these fundamental relationships in mind for more complex problems!

Explore the step-by-step proof of the trigonometric identity cot A = 1 / tan A, based on reciprocal relationships between trigonometric functions. This solution covers key concepts such as the definitions of tangent and cotangent, as well as reciprocal identities, making it suitable for students in Grades 9-12.

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