Evaluate the following expression:
`tan("cosec"^(-1)65/63)`

Explanation

To evaluate the expression \( \tan(\csc^{-1}(\frac{65}{63})) \), we will follow these steps:

### Step 1: Set Up the Equation
Let \( \theta = \csc^{-1}(\frac{65}{63}) \). This implies that \( \csc(\theta) = \frac{65}{63} \).

### Step 2: Understand the Relationship
Recall that \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, we can write:
\[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{63}{65} \]

### Step 3: Use the Pythagorean Identity
Using the Pythagorean identity, we know:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Substituting \( \sin(\theta) \):
\[ \left(\frac{63}{65}\right)^2 + \cos^2(\theta) = 1 \]
Calculating \( \left(\frac{63}{65}\right)^2 \):
\[ \frac{3969}{4225} + \cos^2(\theta) = 1 \]
Thus,
\[ \cos^2(\theta) = 1 - \frac{3969}{4225} = \frac{4225 - 3969}{4225} = \frac{256}{4225} \]
Taking the square root gives us:
\[ \cos(\theta) = \frac{16}{65} \]

### Step 4: Calculate \( \tan(\theta) \)
Now, we can find \( \tan(\theta) \) using the definition:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{63}{65}}{\frac{16}{65}} = \frac{63}{16} \]

### Final Answer
Thus, we have:
\[ \tan(\csc^{-1}(\frac{65}{63})) = \frac{63}{16} \]