Find the value of `(tan25^(@))/(cot65^(@))`.

Answer

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

### Step 1: Understand the relationship between tangent and cotangent
We know that:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we can express \(\cot 65^\circ\) in terms of tangent:
\[ \cot 65^\circ = \frac{1}{\tan 65^\circ} \]

### Step 2: Rewrite the expression
Now, substituting \(\cot 65^\circ\) in the original expression:
\[ \frac{\tan 25^\circ}{\cot 65^\circ} = \tan 25^\circ \cdot \tan 65^\circ \]

### Step 3: Use the complementary angle identity
Since \(25^\circ\) and \(65^\circ\) are complementary angles (i.e., \(25^\circ + 65^\circ = 90^\circ\)), we can use the identity:
\[ \tan(90^\circ - \theta) = \cot \theta \]
This means:
\[ \tan 65^\circ = \cot 25^\circ \]

### Step 4: Substitute back into the expression
Now substituting \(\tan 65^\circ\) back into our expression:
\[ \tan 25^\circ \cdot \tan 65^\circ = \tan 25^\circ \cdot \cot 25^\circ \]

### Step 5: Simplify the expression
We know that:
\[ \tan \theta \cdot \cot \theta = 1 \]
Thus:
\[ \tan 25^\circ \cdot \cot 25^\circ = 1 \]

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

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