Find the value of `4 tan^-1 (1/5) - tan^-1 (1/239) `

Options

A. `pi`
B. `pi/2`
C. `pi/3`
D. `pi/4`

Answer

To solve the expression \( 4 \tan^{-1} \left( \frac{1}{5} \right) - \tan^{-1} \left( \frac{1}{239} \right) \), we will use the following identities:

1. \( 2 \tan^{-1}(x) = \tan^{-1} \left( \frac{2x}{1 - x^2} \right) \)
2. \( \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1} \left( \frac{x - y}{1 + xy} \right) \)

### Step 1: Rewrite \( 4 \tan^{-1} \left( \frac{1}{5} \right) \)

We can express \( 4 \tan^{-1} \left( \frac{1}{5} \right) \) as \( 2 \cdot 2 \tan^{-1} \left( \frac{1}{5} \right) \).

Using the first identity:
\[ 2 \tan^{-1} \left( \frac{1}{5} \right) = \tan^{-1} \left( \frac{2 \cdot \frac{1}{5}}{1 - \left( \frac{1}{5} \right)^2} \right) \]
Calculating the right-hand side:
\[ = \tan^{-1} \left( \frac{\frac{2}{5}}{1 - \frac{1}{25}} \right) = \tan^{-1} \left( \frac{\frac{2}{5}}{\frac{24}{25}} \right) = \tan^{-1} \left( \frac{2 \cdot 25}{5 \cdot 24} \right) = \tan^{-1} \left( \frac{10}{24} \right) = \tan^{-1} \left( \frac{5}{12} \right) \]

### Step 2: Now compute \( 4 \tan^{-1} \left( \frac{1}{5} \right) \)

Using the result from Step 1:
\[ 4 \tan^{-1} \left( \frac{1}{5} \right) = 2 \tan^{-1} \left( \frac{5}{12} \right) \]
Again applying the first identity:
\[ 2 \tan^{-1} \left( \frac{5}{12} \right) = \tan^{-1} \left( \frac{2 \cdot \frac{5}{12}}{1 - \left( \frac{5}{12} \right)^2} \right) \]
Calculating the right-hand side:
\[ = \tan^{-1} \left( \frac{\frac{10}{12}}{1 - \frac{25}{144}} \right) = \tan^{-1} \left( \frac{\frac{5}{6}}{\frac{119}{144}} \right) = \tan^{-1} \left( \frac{5 \cdot 144}{6 \cdot 119} \right) = \tan^{-1} \left( \frac{120}{119} \right) \]

### Step 3: Now compute \( 4 \tan^{-1} \left( \frac{1}{5} \right) - \tan^{-1} \left( \frac{1}{239} \right) \)

Using the second identity:
\[ \tan^{-1} \left( \frac{120}{119} \right) - \tan^{-1} \left( \frac{1}{239} \right) = \tan^{-1} \left( \frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{119} \cdot \frac{1}{239}} \right) \]

Calculating the numerator:
\[ \frac{120}{119} - \frac{1}{239} = \frac{120 \cdot 239 - 1 \cdot 119}{119 \cdot 239} = \frac{28740 - 119}{28441} = \frac{28721}{28441} \]

Calculating the denominator:
\[ 1 + \frac{120}{119} \cdot \frac{1}{239} = 1 + \frac{120}{28441} = \frac{28441 + 120}{28441} = \frac{28561}{28441} \]

Thus, we have:
\[ \tan^{-1} \left( \frac{28721}{28561} \right) \]

### Step 4: Simplifying the result

Since \( 28721 = 28561 + 160 \), we can see that:
\[ \tan^{-1} \left( \frac{28721}{28561} \right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Result

Thus, the value of \( 4 \tan^{-1} \left( \frac{1}{5} \right) - \tan^{-1} \left( \frac{1}{239} \right) \) is:
\[ \frac{\pi}{4} \]