If `3tan^(-1)x +cot^(-1)x=pi`, then x equals to

Options

A. 0
B. 1
C. `-1`
D. `1/2`

Explanation

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

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

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

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 3\tan^{-1}x - \tan^{-1}x + \frac{\pi}{2} = \pi \]
This simplifies to:
\[ 2\tan^{-1}x + \frac{\pi}{2} = \pi \]

### Step 4: Isolate the term with \(\tan^{-1}x\)
Subtract \(\frac{\pi}{2}\) from both sides:
\[ 2\tan^{-1}x = \pi - \frac{\pi}{2} \]
This simplifies to:
\[ 2\tan^{-1}x = \frac{\pi}{2} \]

### Step 5: Divide by 2
Now, divide both sides by 2:
\[ \tan^{-1}x = \frac{\pi}{4} \]

### Step 6: Solve for \(x\)
Taking the tangent of both sides gives:
\[ x = \tan\left(\frac{\pi}{4}\right) \]
Since \(\tan\left(\frac{\pi}{4}\right) = 1\), we find:
\[ x = 1 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{1} \]

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