`cos^(-1)(15/17)+2 tan^(-1)(1/5)=`

Options

A. `(pi)/(2)`
B. `cos^(-1)(171)/(221)`
C. `(pi)/(4)`
D. none of these

Answer

To solve the expression \( \cos^{-1}\left(\frac{15}{17}\right) + 2 \tan^{-1}\left(\frac{1}{5}\right) \), we will follow these steps:

### Step 1: Convert \( 2 \tan^{-1}\left(\frac{1}{5}\right) \) to \( \cos^{-1} \)
We can use the identity:
\[ 2 \tan^{-1}(x) = \cos^{-1\left(\frac{1 - x^2}{1 + x^2}\right)} \]
Here, \( x = \frac{1}{5} \).

### Step 2: Calculate \( 1 - x^2 \) and \( 1 + x^2 \)
Calculating \( 1 - \left(\frac{1}{5}\right)^2 \) and \( 1 + \left(\frac{1}{5}\right)^2 \):
\[ 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{24}{25} \]
\[ 1 + \left(\frac{1}{5}\right)^2 = 1 + \frac{1}{25} = \frac{26}{25} \]

### Step 3: Substitute into the identity
Now we substitute these values into the identity:
\[ 2 \tan^{-1}\left(\frac{1}{5}\right) = \cos^{-1}\left(\frac{\frac{24}{25}}{\frac{26}{25}}\right) = \cos^{-1}\left(\frac{24}{26}\right) = \cos^{-1}\left(\frac{12}{13}\right) \]

### Step 4: Combine the two \( \cos^{-1} \) terms
Now we have:
\[ \cos^{-1}\left(\frac{15}{17}\right) + \cos^{-1}\left(\frac{12}{13}\right) \]
We can use the identity:
\[ \cos^{-1}(x) + \cos^{-1}(y) = \cos^{-1}(xy + \sqrt{(1-x^2)(1-y^2)}) \]

### Step 5: Calculate \( xy \)
Let \( x = \frac{15}{17} \) and \( y = \frac{12}{13} \):
\[ xy = \frac{15}{17} \cdot \frac{12}{13} = \frac{180}{221} \]

### Step 6: Calculate \( \sqrt{(1-x^2)(1-y^2)} \)
Now we calculate \( 1 - x^2 \) and \( 1 - y^2 \):
\[ 1 - \left(\frac{15}{17}\right)^2 = 1 - \frac{225}{289} = \frac{64}{289} \]
\[ 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \]
Now calculate:
\[ \sqrt{(1-x^2)(1-y^2)} = \sqrt{\frac{64}{289} \cdot \frac{25}{169}} = \sqrt{\frac{1600}{48961}} = \frac{40}{221} \]

### Step 7: Combine results
Now we can combine the results:
\[ xy + \sqrt{(1-x^2)(1-y^2)} = \frac{180}{221} + \frac{40}{221} = \frac{220}{221} \]

### Step 8: Final result
Thus, we have:
\[ \cos^{-1}\left(\frac{15}{17}\right) + 2 \tan^{-1}\left(\frac{1}{5}\right) = \cos^{-1}\left(\frac{220}{221}\right) \]

### Conclusion
The final answer is:
\[ \cos^{-1}\left(\frac{220}{221}\right) \]