If `|sin x+cos x|=|sin x|+|cos x| (sin x ,cos x !=0)`, then in which quadrant does `x` lie?

Explanation

To solve the equation \( | \sin x + \cos x | = | \sin x | + | \cos x | \) under the condition that \( \sin x \neq 0 \) and \( \cos x \neq 0 \), we need to analyze the conditions under which this equality holds.

### Step-by-step Solution:

1. **Understanding the Absolute Value Condition**:
The equation \( | \sin x + \cos x | = | \sin x | + | \cos x | \) implies that the expression inside the absolute value on the left must have the same sign as the expressions on the right. This means that both \( \sin x \) and \( \cos x \) must either be both positive or both negative.

2. **Case Analysis**:
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