Evaluate `tan(cosec^(-1)(5/3))`.

Answer

To evaluate \( \tan(\csc^{-1}(5/3)) \), we can follow these steps:

### Step 1: Set up the equation
Let \( \theta = \csc^{-1}(5/3) \). This implies that \( \csc(\theta) = \frac{5}{3} \).

### Step 2: Relate cosecant to sine
Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we can write:
\[ \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{3}{5} \]

### Step 3: Use the Pythagorean identity
In a right triangle, if we know the sine of an angle, we can find the other sides. Here, we can consider:
- Hypotenuse = 5 (from \( \csc(\theta) \))
- Opposite side (perpendicular) = 3 (from \( \sin(\theta) \))

To find the adjacent side (base), we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + \text{Adjacent}^2 \]
\[ 25 = 9 + \text{Adjacent}^2 \]
\[ \text{Adjacent}^2 = 25 - 9 = 16 \]
\[ \text{Adjacent} = \sqrt{16} = 4 \]

### Step 4: Calculate tangent
Now that we have both the opposite side and the adjacent side, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{4} \]

### Step 5: Conclusion
Thus, we find that:
\[ \tan(\csc^{-1}(5/3)) = \frac{3}{4} \]