How many solutions does the equation `5 tan^-1 x+3 cot^-1 x=2pi` have ?

Options

A. zero
B. Exactly one
C. Exactly two
D. Infinite

Answer

To solve the equation \( 5 \tan^{-1} x + 3 \cot^{-1} x = 2\pi \), we can follow these steps:

### Step 1: Use the identity for \(\cot^{-1} x\)
We know that:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Thus, we can rewrite the equation:
\[ 5 \tan^{-1} x + 3 \left( \frac{\pi}{2} - \tan^{-1} x \right) = 2\pi \]

### Step 2: Simplify the equation
Substituting the identity into the equation gives:
\[ 5 \tan^{-1} x + \frac{3\pi}{2} - 3 \tan^{-1} x = 2\pi \]
Combining like terms results in:
\[ (5 - 3) \tan^{-1} x + \frac{3\pi}{2} = 2\pi \]
This simplifies to:
\[ 2 \tan^{-1} x + \frac{3\pi}{2} = 2\pi \]

### Step 3: Isolate \(\tan^{-1} x\)
Now, we can isolate \(\tan^{-1} x\):
\[ 2 \tan^{-1} x = 2\pi - \frac{3\pi}{2} \]
Calculating the right-hand side:
\[ 2 \tan^{-1} x = 2\pi - \frac{3\pi}{2} = \frac{4\pi}{2} - \frac{3\pi}{2} = \frac{\pi}{2} \]

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

### Step 5: Find the value of \(x\)
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:
\[ x = 1 \]

### Step 6: Determine the number of solutions
The equation \(5 \tan^{-1} x + 3 \cot^{-1} x = 2\pi\) has exactly one solution, which is \(x = 1\).

### Conclusion
Thus, the number of solutions to the equation is **1**.

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