The general solution of `|sinx|=cosx` is (When `ninz`) given by

Options

A. `2npi+(pi)/4,n epsilonI`
B. `npi+(pi)/4,n epsilonI`
C. `2npi+-(pi)/4epsilonI`
D. `npi+-(pi)/4,n epsilonI`

Explanation

To solve the equation \( |\sin x| = \cos x \), we will follow these steps:

### Step 1: Square both sides
We start with the equation:
\[ |\sin x| = \cos x \]
Squaring both sides gives us:
\[ \sin^2 x = \cos^2 x \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting \(\sin^2 x\) with \(1 - \cos^2 x\) in our previous equation:
\[ 1 - \cos^2 x = \cos^2 x \]

### Step 3: Rearranging the equation
Rearranging the equation gives us:
\[ 1 = 2\cos^2 x \]
Dividing both sides by 2:
\[ \cos^2 x = \frac{1}{2} \]

### Step 4: Taking the square root
Taking the square root of both sides, we find:
\[ \cos x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Step 5: Finding angles
The cosine function equals \(\frac{\sqrt{2}}{2}\) at angles:
\[ x = \frac{\pi}{4} + 2n\pi \quad \text{and} \quad x = -\frac{\pi}{4} + 2n\pi \]
The cosine function equals \(-\frac{\sqrt{2}}{2}\) at angles:
\[ x = \frac{3\pi}{4} + 2n\pi \quad \text{and} \quad x = -\frac{3\pi}{4} + 2n\pi \]

### Step 6: Combining the solutions
Thus, the general solutions can be combined as:
\[ x = 2n\pi + \frac{\pi}{4} \quad \text{or} \quad x = 2n\pi - \frac{\pi}{4} \]
This can be written as:
\[ x = 2n\pi \pm \frac{\pi}{4} \]

### Final Answer
The general solution of \( |\sin x| = \cos x \) is:
\[ x = 2n\pi \pm \frac{\pi}{4}, \quad n \in \mathbb{Z} \]

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