Number of solutions of the equation `cot^(-1)sqrt(4-x^(2))+cos^(-1)(x^(2)-5)=(3pi)/2` is:

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the equation \( \cot^{-1}(\sqrt{4 - x^2}) + \cos^{-1}(x^2 - 5) = \frac{3\pi}{2} \), we will follow these steps:

### Step 1: Determine the domain of the functions involved

1. **For \( \cot^{-1}(\sqrt{4 - x^2}) \)**:
- The expression \( \sqrt{4 - x^2} \) is defined when \( 4 - x^2 \geq 0 \).
- This implies \( x^2 \leq 4 \) or \( -2 \leq x \leq 2 \).

2. **For \( \cos^{-1}(x^2 - 5) \)**:
- The argument \( x^2 - 5 \) must lie within the range \([-1, 1]\).
- Thus, we have:
\[ -1 \leq x^2 - 5 \leq 1 \]
- This can be split into two inequalities:
1. \( x^2 - 5 \geq -1 \) gives \( x^2 \geq 4 \) or \( |x| \geq 2 \).
2. \( x^2 - 5 \leq 1 \) gives \( x^2 \leq 6 \) or \( |x| \leq \sqrt{6} \).

### Step 2: Combine the domains

- From \( -2 \leq x \leq 2 \) and \( |x| \geq 2 \) (which gives \( x \leq -2 \) or \( x \geq 2 \)), we find that the valid values of \( x \) are:
- \( x = -2 \)
- \( x = 2 \)

### Step 3: Evaluate the equation at the boundary points

1. **For \( x = -2 \)**:
- Calculate \( \cot^{-1}(\sqrt{4 - (-2)^2}) = \cot^{-1}(0) = \frac{\pi}{2} \).
- Calculate \( \cos^{-1}((-2)^2 - 5) = \cos^{-1}(4 - 5) = \cos^{-1}(-1) = \pi \).
- Thus, \( \frac{\pi}{2} + \pi = \frac{3\pi}{2} \).

2. **For \( x = 2 \)**:
- Calculate \( \cot^{-1}(\sqrt{4 - 2^2}) = \cot^{-1}(0) = \frac{\pi}{2} \).
- Calculate \( \cos^{-1}(2^2 - 5) = \cos^{-1}(4 - 5) = \cos^{-1}(-1) = \pi \).
- Thus, \( \frac{\pi}{2} + \pi = \frac{3\pi}{2} \).

### Conclusion

Both \( x = -2 \) and \( x = 2 \) satisfy the equation. Therefore, the number of solutions to the equation is:

\[ \boxed{2} \]