`4 tan^(-1)(1/5)-tan^(-1)(1/239)` is equal to

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 properties and formulas of inverse tangent functions.

### Step 1: Use the double angle formula for tangent
We start by rewriting \( 4 \tan^{-1}\left(\frac{1}{5}\right) \) using the double angle formula:
\[ 4 \tan^{-1}(x) = 2 \tan^{-1}\left(\frac{2x}{1-x^2}\right) \]
Let \( x = \frac{1}{5} \). Then,
\[ 2 \tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2}\right) \]

### Step 2: Calculate the values
Calculating the components:
\[ 2 \cdot \frac{1}{5} = \frac{2}{5} \]
\[ 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{24}{25} \]
Now substituting these back into the formula:
\[ 2 \tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{\frac{2}{5}}{\frac{24}{25}}\right) = \tan^{-1}\left(\frac{2 \cdot 25}{5 \cdot 24}\right) = \tan^{-1}\left(\frac{10}{24}\right) = \tan^{-1}\left(\frac{5}{12}\right) \]
Thus,
\[ 4 \tan^{-1}\left(\frac{1}{5}\right) = 2 \tan^{-1}\left(\frac{5}{12}\right) \]

### Step 3: Substitute back into the original expression
Now we have:
\[ 4 \tan^{-1}\left(\frac{1}{5}\right) - \tan^{-1}\left(\frac{1}{239}\right) = 2 \tan^{-1}\left(\frac{5}{12}\right) - \tan^{-1}\left(\frac{1}{239}\right) \]

### Step 4: Use the tangent subtraction formula
Using the formula for the difference of two arctangents:
\[ \tan^{-1}(x) - \tan^{-1}(y) = \tan^{-1}\left(\frac{x - y}{1 + xy}\right) \]
Let \( x = \frac{5}{12} \) and \( y = \frac{1}{239} \):
\[ 2 \tan^{-1}\left(\frac{5}{12}\right) = \tan^{-1}\left(\frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2}\right) \]

### Step 5: Calculate the components again
Calculating:
\[ 2 \cdot \frac{5}{12} = \frac{10}{12} = \frac{5}{6} \]
\[ 1 - \left(\frac{5}{12}\right)^2 = 1 - \frac{25}{144} = \frac{119}{144} \]
Now substituting back:
\[ 2 \tan^{-1}\left(\frac{5}{12}\right) = \tan^{-1}\left(\frac{\frac{5}{6}}{\frac{119}{144}}\right) = \tan^{-1}\left(\frac{5 \cdot 144}{6 \cdot 119}\right) = \tan^{-1}\left(\frac{120}{119}\right) \]

### Step 6: Final substitution
Now we have:
\[ \tan^{-1}\left(\frac{120}{119}\right) - \tan^{-1}\left(\frac{1}{239}\right) \]
Using the subtraction formula:
\[ \tan^{-1}\left(\frac{\frac{120}{119} - \frac{1}{239}}{1 + \frac{120}{119} \cdot \frac{1}{239}}\right) \]

### Step 7: Calculate the numerator and denominator
Calculating the numerator:
\[ \frac{120}{119} - \frac{1}{239} = \frac{120 \cdot 239 - 119}{119 \cdot 239} = \frac{28680 - 119}{28441} = \frac{28561}{28441} \]
Calculating the denominator:
\[ 1 + \frac{120}{119} \cdot \frac{1}{239} = 1 + \frac{120}{28441} = \frac{28441 + 120}{28441} = \frac{28561}{28441} \]

### Step 8: Final result
Thus, we have:
\[ \tan^{-1}\left(\frac{28561}{28561}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

Therefore, the final answer is:
\[ \boxed{\frac{\pi}{4}} \]