The value of ` (2 tan ^(-1) ((3)/(5)) + sin ^(-1) (( 5)/( 13))) ` is equal to:

Options

A. `(151)/(63)`
B. `(220)/(21)`
C. `(-181)/(69)`
D. `(-291)/(76)`

Answer

To solve the expression \( 2 \tan^{-1} \left( \frac{3}{5} \right) + \sin^{-1} \left( \frac{5}{13} \right) \), we can follow these steps:

### Step 1: Set Variables
Let:
\[ \theta = \tan^{-1} \left( \frac{3}{5} \right) \]
\[ \alpha = \sin^{-1} \left( \frac{5}{13} \right) \]

### Step 2: Find \( \tan(2\theta) \)
Using the double angle formula for tangent:
\[ \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \]
First, we need to find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{3}{5} \]
Now, calculate \( \tan^2(\theta) \):
\[ \tan^2(\theta) = \left( \frac{3}{5} \right)^2 = \frac{9}{25} \]
Now substitute into the double angle formula:
\[ \tan(2\theta) = \frac{2 \cdot \frac{3}{5}}{1 - \frac{9}{25}} = \frac{\frac{6}{5}}{1 - \frac{9}{25}} = \frac{\frac{6}{5}}{\frac{16}{25}} = \frac{6}{5} \cdot \frac{25}{16} = \frac{30}{16} = \frac{15}{8} \]

### Step 3: Find \( \tan(\alpha) \)
From \( \alpha = \sin^{-1} \left( \frac{5}{13} \right) \), we know:
\[ \sin(\alpha) = \frac{5}{13} \]
Using the Pythagorean theorem to find \( \cos(\alpha) \):
\[ \cos(\alpha) = \sqrt{1 - \sin^2(\alpha)} = \sqrt{1 - \left( \frac{5}{13} \right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \]
Now, calculate \( \tan(\alpha) \):
\[ \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{5/13}{12/13} = \frac{5}{12} \]

### Step 4: Use the Formula for \( \tan(2\theta + \alpha) \)
Now we can use the formula:
\[ \tan(2\theta + \alpha) = \frac{\tan(2\theta) + \tan(\alpha)}{1 - \tan(2\theta) \tan(\alpha)} \]
Substituting the values:
\[ \tan(2\theta + \alpha) = \frac{\frac{15}{8} + \frac{5}{12}}{1 - \frac{15}{8} \cdot \frac{5}{12}} \]

### Step 5: Find a Common Denominator
Calculate the numerator:
\[ \frac{15}{8} + \frac{5}{12} = \frac{15 \cdot 3}{24} + \frac{5 \cdot 2}{24} = \frac{45 + 10}{24} = \frac{55}{24} \]
Now calculate the denominator:
\[ 1 - \frac{15}{8} \cdot \frac{5}{12} = 1 - \frac{75}{96} = \frac{96 - 75}{96} = \frac{21}{96} \]

### Step 6: Final Calculation
Now substitute back into the formula:
\[ \tan(2\theta + \alpha) = \frac{\frac{55}{24}}{\frac{21}{96}} = \frac{55}{24} \cdot \frac{96}{21} = \frac{55 \cdot 4}{21} = \frac{220}{21} \]

### Conclusion
Thus, the value of \( 2 \tan^{-1} \left( \frac{3}{5} \right) + \sin^{-1} \left( \frac{5}{13} \right) \) is:
\[ \frac{220}{21} \]