The number of real solutions of the equation
`sqrt(1+cos2x) = sqrt(2)cos^(-1)(cosx)` in `[pi/2,pi]` is

Options

A. 0
B. 1
C. 2
D. `oo`

Explanation

To solve the equation \( \sqrt{1 + \cos 2x} = \sqrt{2} \cos^{-1}(\cos x) \) in the interval \([ \frac{\pi}{2}, \pi ]\), we will follow these steps:

### Step 1: Simplify the left-hand side
We know that:
\[ \cos 2x = 2 \cos^2 x - 1 \]
Thus,
\[ 1 + \cos 2x = 1 + (2 \cos^2 x - 1) = 2 \cos^2 x \]
Taking the square root gives:
\[ \sqrt{1 + \cos 2x} = \sqrt{2 \cos^2 x} = \sqrt{2} |\cos x| \]

### Step 2: Simplify the right-hand side
The right-hand side is:
\[ \sqrt{2} \cos^{-1}(\cos x) \]
In the interval \([ \frac{\pi}{2}, \pi ]\), \( \cos x \) is non-positive, and \( \cos^{-1}(\cos x) = x \) when \( x \) is in this range. Therefore:
\[ \sqrt{2} \cos^{-1}(\cos x) = \sqrt{2} x \]

### Step 3: Set the two sides equal
Now we have:
\[ \sqrt{2} |\cos x| = \sqrt{2} x \]
Dividing both sides by \( \sqrt{2} \) (which is positive) gives:
\[ |\cos x| = x \]

### Step 4: Analyze the equation
In the interval \([ \frac{\pi}{2}, \pi ]\):
- \( \cos x \) is non-positive, so \( |\cos x| = -\cos x \).
Thus, we rewrite the equation as:
\[ -\cos x = x \]
or
\[ \cos x = -x \]

### Step 5: Find intersections
We need to find the number of solutions to the equation \( \cos x = -x \) in the interval \([ \frac{\pi}{2}, \pi ]\).

1. The function \( \cos x \) decreases from \( 0 \) to \( -1 \) as \( x \) goes from \( \frac{\pi}{2} \) to \( \pi \).
2. The function \( -x \) is a straight line that decreases from \( -\frac{\pi}{2} \) to \( -\pi \).

### Step 6: Graphical interpretation
- At \( x = \frac{\pi}{2} \), \( \cos(\frac{\pi}{2}) = 0 \) and \( -\frac{\pi}{2} < 0 \).
- At \( x = \pi \), \( \cos(\pi) = -1 \) and \( -\pi < -1 \).

Since \( \cos x \) starts at \( 0 \) and goes to \( -1 \) while \( -x \) starts at \( -\frac{\pi}{2} \) and goes to \( -\pi \), these two functions will intersect exactly once in the interval \([ \frac{\pi}{2}, \pi ]\).

### Conclusion
Thus, the number of real solutions of the equation in the given interval is:
\[ \text{Number of solutions} = 1 \]