Solve `3 tan^(-1) x+cot^(-1)x=pi`. a) 0 b) 1 c) -1 d) 1/2

Options

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

Answer

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:
\[ 3 \tan^{-1} x + \cot^{-1} x = \pi \]
We can express \( \cot^{-1} x \) in terms of \( \tan^{-1} x \):
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Substituting this into the equation gives:
\[ 3 \tan^{-1} x + \left( \frac{\pi}{2} - \tan^{-1} x \right) = \pi \]

### Step 2: 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 3: Isolate \( \tan^{-1} x \)
Next, isolate \( 2 \tan^{-1} x \):
\[ 2 \tan^{-1} x = \pi - \frac{\pi}{2} \]
This simplifies to:
\[ 2 \tan^{-1} x = \frac{\pi}{2} \]

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

### Step 5: Find \( x \)
Taking the tangent of both sides gives:
\[ x = \tan\left(\frac{\pi}{4}\right) \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \):
\[ x = 1 \]

### Final Answer
Thus, the solution to the equation \( 3 \tan^{-1} x + \cot^{-1} x = \pi \) is:
\[ \boxed{1} \]