The exhaustive set of values of a for which `a - cot^(-1) 3x = 2 tan^(-1) 3x + cos^(-1) x sqrt3 + sin^(-1) x sqrt3` may have solution, is

Options

A. `[-(pi)/(4), (pi)/(4)]`
B. `((pi)/(2), (3pi)/(2))`
C. `[(2pi)/(3), (4pi)/(3)]`
D. `[-(3pi)/(6), (7pi)/(6)]`

Explanation

To solve the equation \( a - \cot^{-1}(3x) = 2\tan^{-1}(3x) + \cos^{-1}(x\sqrt{3}) + \sin^{-1}(x\sqrt{3}) \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ a - \cot^{-1}(3x) = 2\tan^{-1}(3x) + \cos^{-1}(x\sqrt{3}) + \sin^{-1}(x\sqrt{3}) \]

### Step 2: Use the identity for \(\cos^{-1}\) and \(\sin^{-1}\)
Recall that:
\[ \cos^{-1}(y) + \sin^{-1}(y) = \frac{\pi}{2} \]
for \(y = x\sqrt{3}\). Therefore, we can rewrite:
\[ \cos^{-1}(x\sqrt{3}) + \sin^{-1}(x\sqrt{3}) = \frac{\pi}{2} \]

### Step 3: Substitute back into the equation
Substituting this back into our equation gives:
\[ a - \cot^{-1}(3x) = 2\tan^{-1}(3x) + \frac{\pi}{2} \]

### Step 4: Rearranging the equation
Rearranging the equation yields:
\[ a = \cot^{-1}(3x) + 2\tan^{-1}(3x) + \frac{\pi}{2} \]

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

### Step 6: Determine the range of \(a\)
Now, we need to find the range of \(a\) as \(x\) varies. The function \(\tan^{-1}(3x)\) varies from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) as \(x\) varies from \(-\infty\) to \(+\infty\). Therefore, we can find the limits for \(a\):
- When \(x \to -\infty\), \(\tan^{-1}(3x) \to -\frac{\pi}{2}\), thus:
\[ a \to \pi - \frac{\pi}{2} = \frac{\pi}{2} \]
- When \(x \to +\infty\), \(\tan^{-1}(3x) \to \frac{\pi}{2}\), thus:
\[ a \to \pi + \frac{\pi}{2} = \frac{3\pi}{2} \]

### Step 7: Conclusion
The exhaustive set of values of \(a\) for which the equation has solutions is:
\[ \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \]