If `0 lt x lt 2 pi` and `|cos x| le sin x`, then

Options

A. the set of all values of x is `[pi/4, (3pi)/4]`
B. the number of solutions that are integral multiple of `pi/2` is four
C. the sum of the largest and the smallest solution is `pi`
D. the set of all values of x is `x in [pi/4, pi/2) uu (pi/2, (3pi)/4]`

Explanation

To solve the problem where \( | \cos x | \leq \sin x \) for \( 0 < x < 2\pi \), we will analyze the inequality step by step.

### Step 1: Understanding the Inequality
The inequality \( | \cos x | \leq \sin x \) can be interpreted as two separate inequalities:
1. \( \cos x \leq \sin x \)
2. \( -\cos x \leq \sin x \) (which is equivalent to \( \cos x \geq -\sin x \))

### Step 2: Solve the First Inequality \( \cos x \leq \sin x \)
To solve \( \cos x \leq \sin x \), we can rearrange it:
\[ \cos x - \sin x \leq 0 \]
This can be rewritten as:
\[ \tan x \geq 1 \]
This inequality holds true when:
\[ x \in \left[\frac{\pi}{4} + n\pi, \frac{5\pi}{4} + n\pi\right] \quad \text{for integers } n \]
Within the interval \( 0 < x < 2\pi \), this gives us:
\[ x \in \left[\frac{\pi}{4}, \frac{5\pi}{4}\right] \]

### Step 3: Solve the Second Inequality \( -\cos x \leq \sin x \)
Rearranging gives:
\[ \cos x + \sin x \geq 0 \]
This can be rewritten as:
\[ \tan x \geq -1 \]
This inequality holds true when:
\[ x \in \left[-\frac{\pi}{4} + n\pi, \frac{3\pi}{4} + n\pi\right] \quad \text{for integers } n \]
Within the interval \( 0 < x < 2\pi \), this gives us:
\[ x \in \left[\frac{3\pi}{4}, \frac{7\pi}{4}\right] \]

### Step 4: Find the Intersection of the Two Intervals
Now we need to find the intersection of the two intervals:
1. From \( \cos x \leq \sin x \): \( x \in \left[\frac{\pi}{4}, \frac{5\pi}{4}\right] \)
2. From \( -\cos x \leq \sin x \): \( x \in \left[\frac{3\pi}{4}, \frac{7\pi}{4}\right] \)

The intersection of these intervals is:
\[ x \in \left[\frac{3\pi}{4}, \frac{5\pi}{4}\right] \]

### Final Answer
Thus, the solution to the inequality \( | \cos x | \leq \sin x \) for \( 0 < x < 2\pi \) is:
\[ x \in \left[\frac{3\pi}{4}, \frac{5\pi}{4}\right] \]