`tan^(-1)(1/2)+tan^(-1)(1/3)=?`

Options

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

Explanation

To solve the expression \( \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) \), we can use the formula for the sum of two inverse tangent functions:

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

provided that \( xy < 1 \).

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

### Step 2: Check the condition \( xy < 1 \)
Calculate \( xy \):
\[ xy = \left(\frac{1}{2}\right) \left(\frac{1}{3}\right) = \frac{1}{6} \]
Since \( \frac{1}{6} < 1 \), we can apply the formula.

### Step 3: Apply the formula
Using the formula:
\[ \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1}\left(\frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{6}}\right) \]

### Step 4: Simplify the numerator and denominator
Calculate the numerator:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

Calculate the denominator:
\[ 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

### Step 5: Substitute back into the formula
Now substitute the simplified numerator and denominator back into the formula:
\[ \tan^{-1}\left(\frac{\frac{5}{6}}{\frac{5}{6}}\right) = \tan^{-1}(1) \]

### Step 6: Find the value of \( \tan^{-1}(1) \)
We know that:
\[ \tan^{-1}(1) = \frac{\pi}{4} \]

### Final Answer
Thus, the value of \( \tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{3}\right) \) is:
\[ \frac{\pi}{4} \]

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