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

Options

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

Answer

To solve the equation \( 3 \tan^{-1} x + \cot^{-1} x = \pi \), we can 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:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Substituting this into our equation gives:
\[ 3 \tan^{-1} x + \left(\frac{\pi}{2} - \tan^{-1} x\right) = \pi \]

### Step 3: Simplify the equation
Now, we can 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 \( \tan^{-1} x \)
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, we have:
\[ x = \tan\left(\frac{\pi}{4}\right) \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we find:
\[ x = 1 \]

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