Find the value of :
`4(2 tan^(-1) . 1/3 +tan^(-1). 1/7)`

Answer

To find the value of \( 4 \left( 2 \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) \right) \), we will follow these steps:

### Step 1: Simplify the expression inside the brackets
We start with the expression:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) \]

### Step 2: Use the double angle formula for tangent
Recall that:
\[ 2 \tan^{-1}(x) = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \]
Let \( x = \frac{1}{3} \). Then:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{2 \cdot \frac{1}{3}}{1 - \left( \frac{1}{3} \right)^2} \right) \]
Calculating the denominator:
\[ 1 - \left( \frac{1}{3} \right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]
Now substituting back:
\[ 2 \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{\frac{2}{3}}{\frac{8}{9}} \right) = \tan^{-1} \left( \frac{2 \cdot 9}{3 \cdot 8} \right) = \tan^{-1} \left( \frac{6}{8} \right) = \tan^{-1} \left( \frac{3}{4} \right) \]

### Step 3: Combine the two arctangent terms
Now we have:
\[ \tan^{-1} \left( \frac{3}{4} \right) + \tan^{-1} \left( \frac{1}{7} \right) \]
Using the formula for the sum of arctangents:
\[ \tan^{-1}(A) + \tan^{-1}(B) = \tan^{-1} \left( \frac{A + B}{1 - AB} \right) \]
Let \( A = \frac{3}{4} \) and \( B = \frac{1}{7} \):
\[ AB = \frac{3}{4} \cdot \frac{1}{7} = \frac{3}{28} \]
Now substituting into the formula:
\[ \tan^{-1} \left( \frac{\frac{3}{4} + \frac{1}{7}}{1 - \frac{3}{28}} \right) \]

### Step 4: Calculate the numerator and denominator
Numerator:
\[ \frac{3}{4} + \frac{1}{7} = \frac{21 + 4}{28} = \frac{25}{28} \]
Denominator:
\[ 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]
Thus, we have:
\[ \tan^{-1} \left( \frac{\frac{25}{28}}{\frac{25}{28}} \right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 5: Multiply by 4
Now we return to our original expression:
\[ 4 \left( 2 \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{1}{7} \right) \right) = 4 \cdot \frac{\pi}{4} = \pi \]

### Final Answer
Thus, the value is:
\[ \pi \]