`tan^(-1)((1)/(4))+tan^(-1)((2)/(9))` is equal to :

Options

A. `(1)/(2) cos^(-1)((3)/(5))`
B. `(1)/(2) sin^(-1)((3)/(5))`
C. `(1)/(2) tan^(-1)((3)/(5))`
D. `tan^(-1)((1)/(2))`

Answer

To solve the expression \( \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) \), we can use the formula for the sum of two inverse tangent functions:

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

This formula is valid when \( xy < 1 \).

### Step 1: Identify \( x \) and \( y \)
Let:
- \( x = \frac{1}{4} \)
- \( y = \frac{2}{9} \)

### Step 2: Check the condition \( xy < 1 \)
Calculate \( xy \):
\[ xy = \left(\frac{1}{4}\right) \left(\frac{2}{9}\right) = \frac{2}{36} = \frac{1}{18} \]
Since \( \frac{1}{18} < 1 \), we can apply the formula.

### Step 3: Apply the formula
Now, we can apply the formula:
\[ \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) = \tan^{-1}\left(\frac{\frac{1}{4} + \frac{2}{9}}{1 - \frac{1}{4} \cdot \frac{2}{9}}\right) \]

### Step 4: Calculate the numerator
Calculate \( \frac{1}{4} + \frac{2}{9} \):
To add these fractions, find a common denominator:
- The least common multiple of 4 and 9 is 36.

Convert each fraction:
\[ \frac{1}{4} = \frac{9}{36}, \quad \frac{2}{9} = \frac{8}{36} \]
Now add them:
\[ \frac{1}{4} + \frac{2}{9} = \frac{9}{36} + \frac{8}{36} = \frac{17}{36} \]

### Step 5: Calculate the denominator
Calculate \( 1 - xy \):
\[ 1 - xy = 1 - \frac{1}{18} = \frac{18}{18} - \frac{1}{18} = \frac{17}{18} \]

### Step 6: Combine the results
Now substitute the results back into the formula:
\[ \tan^{-1}\left(\frac{\frac{17}{36}}{\frac{17}{18}}\right) \]

### Step 7: Simplify the fraction
Simplifying the fraction:
\[ \frac{\frac{17}{36}}{\frac{17}{18}} = \frac{17}{36} \cdot \frac{18}{17} = \frac{18}{36} = \frac{1}{2} \]

### Final Result
Thus, we have:
\[ \tan^{-1}\left(\frac{1}{4}\right) + \tan^{-1}\left(\frac{2}{9}\right) = \tan^{-1}\left(\frac{1}{2}\right) \]

### Conclusion
The final answer is:
\[ \tan^{-1}\left(\frac{1}{2}\right) \]