Given that `tan A = (1)/(5).` find the values of tan 2 A, tan 4A and ` tan (45 ^(@) - 4A).`

Answer

To solve the problem step by step, we will find the values of \( \tan 2A \), \( \tan 4A \), and \( \tan(45^\circ - 4A) \) given that \( \tan A = \frac{1}{5} \).

### Step 1: Find \( \tan 2A \)

The formula for \( \tan 2A \) is given by:

\[ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} \]

Substituting \( \tan A = \frac{1}{5} \):

\[ \tan 2A = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} \]

Calculating \( \tan^2 A \):

\[ \tan^2 A = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \]

Now substituting back into the equation:

\[ \tan 2A = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2}{5} \cdot \frac{25}{24} = \frac{10}{24} = \frac{5}{12} \]

### Step 2: Find \( \tan 4A \)

Now we will use the formula for \( \tan 4A \):

\[ \tan 4A = \frac{2 \tan 2A}{1 - \tan^2 2A} \]

Substituting \( \tan 2A = \frac{5}{12} \):

\[ \tan 4A = \frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2} \]

Calculating \( \tan^2 2A \):

\[ \tan^2 2A = \left(\frac{5}{12}\right)^2 = \frac{25}{144} \]

Now substituting back into the equation:

\[ \tan 4A = \frac{\frac{10}{12}}{1 - \frac{25}{144}} = \frac{\frac{10}{12}}{\frac{119}{144}} = \frac{10}{12} \cdot \frac{144}{119} = \frac{120}{119} \]

### Step 3: Find \( \tan(45^\circ - 4A) \)

Using the formula for \( \tan(45^\circ - x) \):

\[ \tan(45^\circ - x) = \frac{1 - \tan x}{1 + \tan x} \]

Substituting \( x = 4A \) and \( \tan 4A = \frac{120}{119} \):

\[ \tan(45^\circ - 4A) = \frac{1 - \frac{120}{119}}{1 + \frac{120}{119}} \]

Calculating the numerator and denominator:

\[ 1 - \frac{120}{119} = \frac{119 - 120}{119} = \frac{-1}{119} \]
\[ 1 + \frac{120}{119} = \frac{119 + 120}{119} = \frac{239}{119} \]

Now substituting back into the equation:

\[ \tan(45^\circ - 4A) = \frac{\frac{-1}{119}}{\frac{239}{119}} = \frac{-1}{239} \]

### Final Answers

1. \( \tan 2A = \frac{5}{12} \)
2. \( \tan 4A = \frac{120}{119} \)
3. \( \tan(45^\circ - 4A) = \frac{-1}{239} \)