`tan^(-1)""1/7+2tan^(-1)"" 1/3=`

Options

A. `-pi//4`
B. `pi//4`
C. `pi//2`
D. `-pi//2`

Answer

To solve the equation \( \tan^{-1}\left(\frac{1}{7}\right) + 2\tan^{-1}\left(\frac{1}{3}\right) \), we will use the properties of inverse tangent functions.

### Step 1: Use the double angle formula for tangent
The double angle formula for tangent states that:
\[ 2\tan^{-1}(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \]
Let \( x = \frac{1}{3} \). Then, we can calculate \( 2\tan^{-1}\left(\frac{1}{3}\right) \):
\[ 2\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{1}{3}}{1 - \left(\frac{1}{3}\right)^2}\right) \]

### Step 2: Substitute and simplify
Calculating the numerator:
\[ 2 \cdot \frac{1}{3} = \frac{2}{3} \]
Calculating the denominator:
\[ 1 - \left(\frac{1}{3}\right)^2 = 1 - \frac{1}{9} = \frac{8}{9} \]
Now substituting these into the formula:
\[ 2\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{\frac{2}{3}}{\frac{8}{9}}\right) = \tan^{-1}\left(\frac{2 \cdot 9}{3 \cdot 8}\right) = \tan^{-1}\left(\frac{6}{8}\right) = \tan^{-1}\left(\frac{3}{4}\right) \]

### Step 3: Combine the angles
Now we need to combine \( \tan^{-1}\left(\frac{1}{7}\right) \) and \( \tan^{-1}\left(\frac{3}{4}\right) \) using the addition formula:
\[ \tan^{-1}(x) + \tan^{-1}(y) = \tan^{-1}\left(\frac{x + y}{1 - xy}\right) \]
Let \( x = \frac{1}{7} \) and \( y = \frac{3}{4} \):
\[ \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{3}{4}\right) = \tan^{-1}\left(\frac{\frac{1}{7} + \frac{3}{4}}{1 - \frac{1}{7} \cdot \frac{3}{4}}\right) \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:
\[ \frac{1}{7} + \frac{3}{4} = \frac{4 + 21}{28} = \frac{25}{28} \]
Calculating the denominator:
\[ 1 - \frac{1}{7} \cdot \frac{3}{4} = 1 - \frac{3}{28} = \frac{28 - 3}{28} = \frac{25}{28} \]

### Step 5: Combine the results
Now substituting back into the formula:
\[ \tan^{-1}\left(\frac{\frac{25}{28}}{\frac{25}{28}}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Answer
Thus, the final result is:
\[ \tan^{-1}\left(\frac{1}{7}\right) + 2\tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} \]