`cot^(-1)(-1/2)+cot^(-1)(-1/3)` is equal to

Options

A. `(3pi)/4`
B. `(5pi)/4`
C. `(pi)/4`
D. `(-3pi)/4`

Explanation

To solve the expression \( \cot^{-1}(-\frac{1}{2}) + \cot^{-1}(-\frac{1}{3}) \), we can follow these steps:

### Step 1: Use the property of inverse cotangent
We know that:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \]
Using this property, we can rewrite our expression:
\[ \cot^{-1}(-\frac{1}{2}) + \cot^{-1}(-\frac{1}{3}) = \left( \pi - \cot^{-1}(\frac{1}{2}) \right) + \left( \pi - \cot^{-1}(\frac{1}{3}) \right) \]

### Step 2: Simplify the expression
Now, we can simplify the expression:
\[ = 2\pi - \cot^{-1}(\frac{1}{2}) - \cot^{-1}(\frac{1}{3}) \]

### Step 3: Combine the cotangent inverses
Next, we can use the identity:
\[ \cot^{-1}(x) + \cot^{-1}(y) = \cot^{-1}\left(\frac{xy - 1}{x + y}\right) \quad \text{if } xy > 1 \]
In our case, we need to find \( \cot^{-1}(\frac{1}{2}) + \cot^{-1}(\frac{1}{3}) \):
\[ \cot^{-1}(\frac{1}{2}) + \cot^{-1}(\frac{1}{3}) = \cot^{-1}\left(\frac{\frac{1}{2} \cdot \frac{1}{3} - 1}{\frac{1}{2} + \frac{1}{3}}\right) \]

### Step 4: Calculate the numerator and denominator
Calculating the numerator:
\[ \frac{1}{2} \cdot \frac{1}{3} - 1 = \frac{1}{6} - 1 = -\frac{5}{6} \]
Calculating the denominator:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
Thus, we have:
\[ \cot^{-1}(\frac{1}{2}) + \cot^{-1}(\frac{1}{3}) = \cot^{-1}\left(\frac{-\frac{5}{6}}{\frac{5}{6}}\right) = \cot^{-1}(-1) = \frac{3\pi}{4} \]

### Step 5: Substitute back into the expression
Now substituting back into our expression:
\[ 2\pi - \left(\frac{3\pi}{4}\right) = 2\pi - \frac{3\pi}{4} = \frac{8\pi}{4} - \frac{3\pi}{4} = \frac{5\pi}{4} \]

### Final Answer
Thus, the final value of \( \cot^{-1}(-\frac{1}{2}) + \cot^{-1}(-\frac{1}{3}) \) is:
\[ \frac{5\pi}{4} \]