Solve the following
` 5 tan^(-1) x + 3 cot^(-1) x = 2 pi `

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 cotangent
We know that:
\[ \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x \]
Substituting this into the equation gives:
\[ 5 \tan^{-1} x + 3 \left(\frac{\pi}{2} - \tan^{-1} x\right) = 2\pi \]

### Step 2: Simplify the equation
Distributing the \(3\) in the equation:
\[ 5 \tan^{-1} x + \frac{3\pi}{2} - 3 \tan^{-1} x = 2\pi \]
Combine like terms:
\[ (5 - 3) \tan^{-1} x + \frac{3\pi}{2} = 2\pi \]
This simplifies to:
\[ 2 \tan^{-1} x + \frac{3\pi}{2} = 2\pi \]

### Step 3: Isolate \( \tan^{-1} x \)
Subtract \( \frac{3\pi}{2} \) from both sides:
\[ 2 \tan^{-1} x = 2\pi - \frac{3\pi}{2} \]
This simplifies to:
\[ 2 \tan^{-1} x = \frac{4\pi}{2} - \frac{3\pi}{2} = \frac{\pi}{2} \]

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

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

### Final Solution
Thus, the solution to the equation \( 5 \tan^{-1} x + 3 \cot^{-1} x = 2\pi \) is:
\[ \boxed{1} \]