The solution of `tan ""(sin^(-1) ""(3)/(5)+ cot^(-1)"" (3)/(2))` is

Options

A. `(17)/(6)`
B. `(17)/(7)`
C. `(17)/(5)`
D. `(17)/(4)`

Answer

To solve the expression \( \tan \left( \sin^{-1} \left( \frac{3}{5} \right) + \cot^{-1} \left( \frac{3}{2} \right) \right) \), we can follow these steps:

### Step 1: Set up the angles
Let \( \theta = \sin^{-1} \left( \frac{3}{5} \right) \). This means that:
\[ \sin \theta = \frac{3}{5} \]

### Step 2: Find the adjacent side
Using the Pythagorean theorem, we can find the adjacent side of the right triangle where:
- Opposite side = 3
- Hypotenuse = 5

Let the adjacent side be \( x \):
\[ x^2 + 3^2 = 5^2 \implies x^2 + 9 = 25 \implies x^2 = 16 \implies x = 4 \]
Thus, the adjacent side is 4.

### Step 3: Calculate \( \tan \theta \)
Now, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} \]

### Step 4: Set up the second angle
Let \( \phi = \cot^{-1} \left( \frac{3}{2} \right) \). This means that:
\[ \cot \phi = \frac{3}{2} \implies \tan \phi = \frac{1}{\cot \phi} = \frac{2}{3} \]

### Step 5: Use the tangent addition formula
Now we need to find \( \tan(\theta + \phi) \) using the formula:
\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]
Substituting the values we have:
\[ \tan(\theta + \phi) = \frac{\frac{3}{4} + \frac{2}{3}}{1 - \left(\frac{3}{4} \cdot \frac{2}{3}\right)} \]

### Step 6: Simplify the numerator
Finding a common denominator for the numerator:
\[ \frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12} \]

### Step 7: Simplify the denominator
Calculating the denominator:
\[ 1 - \left(\frac{3}{4} \cdot \frac{2}{3}\right) = 1 - \frac{6}{12} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 8: Combine the results
Now substituting back into the tangent formula:
\[ \tan(\theta + \phi) = \frac{\frac{17}{12}}{\frac{1}{2}} = \frac{17}{12} \cdot 2 = \frac{34}{12} = \frac{17}{6} \]

### Final Result
Thus, the solution to the expression \( \tan \left( \sin^{-1} \left( \frac{3}{5} \right) + \cot^{-1} \left( \frac{3}{2} \right) \right) \) is:
\[ \frac{17}{6} \]