If `tan^(-1)x+2cot^(-1)x=(2pi)/(3)` then x =

Options

A. 3
B. `sqrt(3)`
C. `sqrt(2)`
D. `(sqrt(3)-1)/(sqrt(3)+1)`

Explanation

To solve the equation \( \tan^{-1} x + 2 \cot^{-1} x = \frac{2\pi}{3} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan^{-1} x + 2 \cot^{-1} x = \frac{2\pi}{3} \]

### Step 2: Use the identity for cotangent
Recall the identity:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Using this identity, we can rewrite \( \cot^{-1} x \):
\[ \tan^{-1} x + 2 \left( \frac{\pi}{2} - \tan^{-1} x \right) = \frac{2\pi}{3} \]

### Step 3: Simplify the equation
Now, distribute the 2:
\[ \tan^{-1} x + \pi - 2 \tan^{-1} x = \frac{2\pi}{3} \]
Combine like terms:
\[ \pi - \tan^{-1} x = \frac{2\pi}{3} \]

### Step 4: Isolate \( \tan^{-1} x \)
Rearranging gives:
\[ \tan^{-1} x = \pi - \frac{2\pi}{3} \]
Calculating the right side:
\[ \tan^{-1} x = \frac{\pi}{3} \]

### Step 5: Solve for \( x \)
To find \( x \), we take the tangent of both sides:
\[ x = \tan\left(\frac{\pi}{3}\right) \]
Since \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \):
\[ x = \sqrt{3} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\sqrt{3}} \]