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

Options

A. `220/21`
B. `110/21`
C. `55/21`
D. `20/11`

Answer

To find the value of \( \tan(2\tan^{-1}(\frac{3}{5}) + \sin^{-1}(\frac{5}{13})) \), we can follow these steps:

### Step 1: Let \( \theta = \tan^{-1}(\frac{3}{5}) \)
From this, we can deduce that:
\[ \tan(\theta) = \frac{3}{5} \]
Using the identity for \( \tan(2\theta) \):
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Substituting \( \tan(\theta) \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{3}{5}}{1 - \left(\frac{3}{5}\right)^2} \]

### Step 2: Calculate \( 1 - \tan^2(\theta) \)
Calculating \( \tan^2(\theta) \):
\[ \tan^2(\theta) = \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]
Thus,
\[ 1 - \tan^2(\theta) = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \]

### Step 3: Substitute back into \( \tan(2\theta) \)
Now substituting back:
\[ \tan(2\theta) = \frac{2 \cdot \frac{3}{5}}{\frac{16}{25}} = \frac{\frac{6}{5}}{\frac{16}{25}} = \frac{6 \cdot 25}{5 \cdot 16} = \frac{30}{16} = \frac{15}{8} \]

### Step 4: Let \( \alpha = \sin^{-1}(\frac{5}{13}) \)
From this, we have:
\[ \sin(\alpha) = \frac{5}{13} \]
Using the Pythagorean theorem, we find \( \cos(\alpha) \):
\[ \cos(\alpha) = \sqrt{1 - \sin^2(\alpha)} = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \sqrt{1 - \frac{25}{169}} = \sqrt{\frac{144}{169}} = \frac{12}{13} \]

### Step 5: Calculate \( \tan(\alpha) \)
Now, we can find \( \tan(\alpha) \):
\[ \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12} \]

### Step 6: 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:
\[ \tan(2\theta + \alpha) = \frac{\frac{15}{8} + \frac{5}{12}}{1 - \left(\frac{15}{8} \cdot \frac{5}{12}\right)} \]

### Step 7: Find a common denominator and simplify
Finding a common denominator for the numerator:
\[ \frac{15}{8} + \frac{5}{12} = \frac{15 \cdot 3}{24} + \frac{5 \cdot 2}{24} = \frac{45 + 10}{24} = \frac{55}{24} \]

Calculating the denominator:
\[ 1 - \left(\frac{15}{8} \cdot \frac{5}{12}\right) = 1 - \frac{75}{96} = \frac{96 - 75}{96} = \frac{21}{96} \]

### Step 8: Final calculation
Now substituting these back:
\[ \tan(2\theta + \alpha) = \frac{\frac{55}{24}}{\frac{21}{96}} = \frac{55 \cdot 96}{24 \cdot 21} = \frac{55 \cdot 4}{21} = \frac{220}{21} \]

Thus, the final answer is:
\[ \boxed{\frac{220}{21}} \]