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 \( \cot^{-1} x \)
We know that:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Using this identity, we can rewrite the equation as:
\[ \tan^{-1} x + 2\left(\frac{\pi}{2} - \tan^{-1} x\right) = \frac{2\pi}{3} \]

### Step 2: Simplify the Equation
Now, substitute \( \cot^{-1} x \) in the equation:
\[ \tan^{-1} x + \pi - 2\tan^{-1} x = \frac{2\pi}{3} \]
Combine like terms:
\[ \pi - \tan^{-1} x = \frac{2\pi}{3} \]

### Step 3: Isolate \( \tan^{-1} x \)
Subtract \( \pi \) from both sides:
\[ -\tan^{-1} x = \frac{2\pi}{3} - \pi \]
This simplifies to:
\[ -\tan^{-1} x = \frac{2\pi}{3} - \frac{3\pi}{3} = -\frac{\pi}{3} \]
Multiplying both sides by -1 gives:
\[ \tan^{-1} x = \frac{\pi}{3} \]

### Step 4: Solve for \( x \)
Using the definition of the inverse tangent function:
\[ x = \tan\left(\frac{\pi}{3}\right) \]
We know that:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus, we find:
\[ x = \sqrt{3} \]

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

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