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 is the greatest among \( \tan 1 \), \( \tan^2 1 \), \( \cot 1 \), and \( \cot^2 1 \), we can follow these steps:

### Step 1: Understand the values of \( \tan 1 \) and \( \cot 1 \)

First, we need to recognize that \( 1 \) is in radians. We know that:
- \( \tan 1 \) is the tangent of \( 1 \) radian.
- \( \cot 1 = \frac{1}{\tan 1} \).

Since \( \tan 1 \) is greater than \( 0 \) (as \( 1 \) radian is in the first quadrant where tangent is positive), we can conclude that \( \cot 1 \) will be less than \( 1 \).

### Step 2: Compare \( \tan 1 \) and \( \cot 1 \)

Since \( \tan 1 > 0 \) and \( \cot 1 < 1 \), we can establish that:
\[ \tan 1 > \cot 1 \]

### Step 3: Analyze \( \tan^2 1 \)

Next, we consider \( \tan^2 1 \):
- Since \( \tan 1 > 1 \) (because \( \tan(\frac{\pi}{4}) = 1 \) and \( 1 \) radian is greater than \( \frac{\pi}{4} \)), squaring \( \tan 1 \) will yield a value greater than \( \tan 1 \):
\[ \tan^2 1 > \tan 1 \]

### Step 4: Compare \( \cot^2 1 \)

Now, we analyze \( \cot^2 1 \):
- Since \( \cot 1 < 1 \), squaring it will yield a value less than \( \cot 1 \):
\[ \cot^2 1 < \cot 1 < 1 \]

### Step 5: Establish the order of the values

From the comparisons made:
1. \( \tan^2 1 > \tan 1 \)
2. \( \tan 1 > \cot 1 \)
3. \( \cot 1 > \cot^2 1 \)

Thus, we can summarize the order as:
\[ \tan^2 1 > \tan 1 > \cot 1 > \cot^2 1 \]

### Conclusion

The greatest value among \( \tan 1 \), \( \tan^2 1 \), \( \cot 1 \), and \( \cot^2 1 \) is:
\[ \boxed{\tan^2 1} \]