What is `tan^(-1)((1)/(2))+tan^(-1)((1)/(3))` equal to?

Options

A. `(pi)/(2)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(6)`

Answer

To solve the problem \( \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) \), we can use the identity for the sum of inverse tangents:

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

where \( x = \frac{1}{2} \) and \( y = \frac{1}{3} \).

### Step 1: Identify \( x \) and \( y \)
Let:
- \( x = \frac{1}{2} \)
- \( y = \frac{1}{3} \)

### Step 2: Apply the formula
Using the formula, we can substitute \( x \) and \( y \):

\[ \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right)}\right) \]

### Step 3: Simplify the numerator
Calculate the numerator:

\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

### Step 4: Simplify the denominator
Calculate the denominator:

\[ 1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right) = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 5: Combine the results
Now, substituting back into the formula gives:

\[ \tan^{-1}\left(\frac{\frac{5}{6}}{\frac{5}{6}}\right) = \tan^{-1}(1) \]

### Step 6: Find the angle
We know that:

\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Conclusion
Thus, we have:

\[ \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4} \]

The correct option is \( \frac{\pi}{4} \).

---