Which of the following is greatest ?

Options

A. tan 1
B. `tan^(2)1`
C. cot 1
D. `cot^(2) 1`

Explanation

To determine which of the following options is greatest between \( \tan(1) \) and \( \cot(1) \), we can analyze the behavior of the tangent and cotangent functions.

### Step-by-Step Solution:

1. **Understanding the Functions**:
- The tangent function, \( \tan(x) \), is defined as the ratio of the sine and cosine functions:
\[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]
- The cotangent function, \( \cot(x) \), is the reciprocal of the tangent function:
\[ \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \]

2. **Behavior of the Functions**:
- The function \( \tan(x) \) is increasing in the interval \( (0, \frac{\pi}{2}) \).
- The function \( \cot(x) \) is decreasing in the same interval.

3. **Evaluating at \( x = 1 \)**:
- Since \( 1 \) (in radians) is between \( 0 \) and \( \frac{\pi}{2} \), we can conclude:
- \( \tan(1) \) is increasing and therefore will have a value greater than \( \tan(0) = 0 \).
- \( \cot(1) \) is decreasing and therefore will have a value less than \( \cot(0) = \infty \).

4. **Comparing Values**:
- Since \( \tan(1) \) is increasing and \( \cot(1) \) is decreasing, we can analyze their relationship:
- If \( \tan(1) = a \), then \( \cot(1) = \frac{1}{a} \).
- As \( a \) increases, \( \frac{1}{a} \) decreases.

5. **Conclusion**:
- Therefore, \( \tan(1) \) will always be greater than \( \cot(1) \) because:
\[ \tan(1) > \cot(1) \]
- Hence, the greatest value among the options is \( \tan(1) \).

### Final Answer:
The greatest value is \( \tan(1) \).