Evaluate `(tan65^@)/(cot25^@)`.

Options

A. `1`
B. `2`
C. `3`
D. `4`

Explanation

To evaluate the expression \(\frac{\tan 65^\circ}{\cot 25^\circ}\), we can follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We know that \(\cot \theta = \frac{1}{\tan \theta}\). Therefore, we can rewrite \(\cot 25^\circ\) as:
\[ \cot 25^\circ = \frac{1}{\tan 25^\circ} \]
Thus, we can rewrite the expression as:
\[ \frac{\tan 65^\circ}{\cot 25^\circ} = \tan 65^\circ \cdot \tan 25^\circ \]

### Step 2: Use the complementary angle identity
We know that:
\[ \tan(90^\circ - \theta) = \cot \theta \]
This means:
\[ \tan 65^\circ = \cot 25^\circ \]
Thus, we can substitute \(\tan 65^\circ\) with \(\cot 25^\circ\):
\[ \tan 65^\circ = \cot 25^\circ \]

### Step 3: Substitute back into the expression
Now we substitute back into our expression:
\[ \tan 65^\circ \cdot \tan 25^\circ = \cot 25^\circ \cdot \tan 25^\circ \]

### Step 4: Simplify the expression
Since \(\cot 25^\circ = \frac{1}{\tan 25^\circ}\), we have:
\[ \cot 25^\circ \cdot \tan 25^\circ = 1 \]

### Final Answer
Thus, the value of the expression \(\frac{\tan 65^\circ}{\cot 25^\circ}\) is:
\[ \boxed{1} \]