The value of `cot ((pi)/(2) - 2 cot^(-1) sqrt3)` is :

Options

A. 0
B. `(1)/(sqrt3)`
C. `sqrt3`
D. `1`

Answer

To solve the problem, we need to find the value of \( \cot\left(\frac{\pi}{2} - 2 \cot^{-1}(\sqrt{3})\right) \).

### Step 1: Simplify \( 2 \cot^{-1}(\sqrt{3}) \)
First, we need to find \( \cot^{-1}(\sqrt{3}) \). The angle whose cotangent is \( \sqrt{3} \) is \( \frac{\pi}{6} \) (since \( \cot\left(\frac{\pi}{6}\right) = \sqrt{3} \)). Thus,

\[ \cot^{-1}(\sqrt{3}) = \frac{\pi}{6} \]

Now, we calculate \( 2 \cot^{-1}(\sqrt{3}) \):

\[ 2 \cot^{-1}(\sqrt{3}) = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]

### Step 2: Substitute back into the cotangent function
Now we substitute this back into our original expression:

\[ \cot\left(\frac{\pi}{2} - 2 \cot^{-1}(\sqrt{3})\right) = \cot\left(\frac{\pi}{2} - \frac{\pi}{3}\right) \]

### Step 3: Simplify the angle
Now we simplify the angle:

\[ \frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \]

### Step 4: Find the cotangent of the resulting angle
Now we need to find \( \cot\left(\frac{\pi}{6}\right) \):

\[ \cot\left(\frac{\pi}{6}\right) = \frac{1}{\tan\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Final Answer
Thus, the value of \( \cot\left(\frac{\pi}{2} - 2 \cot^{-1}(\sqrt{3})\right) \) is:

\[ \sqrt{3} \]