Evaluate the following :
`tan ( cosec^(-1) sqrt(41)/4)`

Answer

To evaluate \( \tan \left( \csc^{-1} \left( \frac{\sqrt{41}}{4} \right) \right) \), we can follow these steps:

### Step 1: Set up the equation
Let
\[ x = \csc^{-1} \left( \frac{\sqrt{41}}{4} \right) \]
This implies that
\[ \csc x = \frac{\sqrt{41}}{4} \]

### Step 2: Find sine
Since \( \csc x = \frac{1}{\sin x} \), we can write:
\[ \sin x = \frac{1}{\csc x} = \frac{4}{\sqrt{41}} \]

### Step 3: Find cosine
Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), we can find \( \cos x \):
\[ \cos^2 x = 1 - \sin^2 x = 1 - \left( \frac{4}{\sqrt{41}} \right)^2 \]
Calculating \( \sin^2 x \):
\[ \sin^2 x = \frac{16}{41} \]
So,
\[ \cos^2 x = 1 - \frac{16}{41} = \frac{41 - 16}{41} = \frac{25}{41} \]
Thus,
\[ \cos x = \sqrt{\frac{25}{41}} = \frac{5}{\sqrt{41}} \]

### Step 4: Find tangent
Now we can find \( \tan x \):
\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{4}{\sqrt{41}}}{\frac{5}{\sqrt{41}}} \]
This simplifies to:
\[ \tan x = \frac{4}{5} \]

### Final Result
Thus,
\[ \tan \left( \csc^{-1} \left( \frac{\sqrt{41}}{4} \right) \right) = \frac{4}{5} \]

---