Conisder x = 4 `tan^(-1)((1)/(5)),y = tan^(-1)((1)/(70))and z = tan^(-1)((1)/(99))`.
What is x equal to ?

Options

A. `tan^(-1)((60)/(119))`
B. `tan^(-1)((120)/(119))`
C. `tan^(-1)((90)/(169))`
D. `tan^(-1)((170)/(169))`

Explanation

To find the value of \( x \) where \( x = 4 \tan^{-1}\left(\frac{1}{5}\right) \), we will use the properties of the inverse tangent function.

### Step-by-Step Solution:

1. **Start with the given expression:**
\[ x = 4 \tan^{-1}\left(\frac{1}{5}\right) \]

2. **Use the double angle formula for tangent:**
The double angle formula for tangent states that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
Here, let \( \theta = \tan^{-1}\left(\frac{1}{5}\right) \), so \( \tan(\theta) = \frac{1}{5} \).

3. **Calculate \( \tan(2\theta) \):**
Using the formula:
\[ \tan(2\theta) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2 \cdot 25}{5 \cdot 24} = \frac{10}{24} = \frac{5}{12} \]
Therefore, we have:
\[ 2 \tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{5}{12}\right) \]

4. **Now calculate \( x \) as \( 2 \tan^{-1}\left(\frac{5}{12}\right) \):**
\[ x = 2 \tan^{-1}\left(\frac{5}{12}\right) \]

5. **Apply the double angle formula again:**
Now, we can apply the double angle formula again:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]
where \( \theta = \tan^{-1}\left(\frac{5}{12}\right) \).

6. **Calculate \( \tan(2\theta) \):**
\[ \tan(2\theta) = \frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2} = \frac{\frac{10}{12}}{1 - \frac{25}{144}} = \frac{\frac{10}{12}}{\frac{119}{144}} = \frac{10 \cdot 144}{12 \cdot 119} = \frac{120}{119} \]

7. **Final result:**
Thus, we find:
\[ x = \tan^{-1}\left(\frac{120}{119}\right) \]

### Conclusion:
The value of \( x \) is:
\[ x = \tan^{-1}\left(\frac{120}{119}\right) \]