**Finding the Domain of √(cos(sin x))**

To find the domain of the function √(cos(sin x)), we need to determine the values of x for which the function is defined. The domain represents the set of all possible input values that can be plugged into the function.

**Step 1: Consider the Innermost Function**

The innermost function in the given expression is sin x. The domain of sin x is all real numbers, as the sine function is defined for any value of x.

**Step 2: Evaluate the Intermediate Function**

Next, we consider the intermediate function, which is cos(sin x). The range of sin x is between -1 and 1, inclusive. Therefore, the range of cos(sin x) is also between -1 and 1, inclusive.

Since the square root function (√) is only defined for non-negative numbers, we need to find the values of cos(sin x) that fall within this range.

**Step 3: Find the Intersection of Domains**

The domain of √(cos(sin x)) will be the intersection of the domains of the intermediate function cos(sin x) and the square root function √(cos(sin x)).

The square root function is defined for values greater than or equal to zero. Therefore, the domain of √(cos(sin x)) will only include the values of x for which cos(sin x) is greater than or equal to zero.

**Step 4: Determine the Domain of cos(sin x)**

To find the domain of cos(sin x), we consider the range of sin x. Since the range of sin x is between -1 and 1, inclusive, the range of cos(sin x) will also be between -1 and 1, inclusive.

However, since we are looking for the values of x for which cos(sin x) is greater than or equal to zero, we need to identify the intervals where cos(sin x) is positive.

**Step 5: Identify the Intervals where cos(sin x) is Positive**

To determine the intervals where cos(sin x) is positive, we consider the unit circle and the quadrants where cos x is positive. Since sin x is involved in the expression, we need to consider the quadrants where sin x is positive as well.

In the unit circle, sin x is positive in the first and second quadrants, while cos x is positive in the first and fourth quadrants.

Therefore, the intervals where cos(sin x) is positive are:

- First quadrant: 0 ≤ x < />
- Second quadrant: π/2 < x="" />< />

**Step 6: Determine the Intersection of Domains**

Finally, we find the intersection of the domain of cos(sin x) and the domain of the square root function:

- Domain of cos(sin x): 0 ≤ x < π/2="" or="" π/2="" />< x="" />< />
- Domain of √(cos(sin x)): 0 ≤ x < π/2="" or="" π/2="" />< x="" />< />

Thus, the domain of the function √(cos(sin x)) is 0 ≤ x < π/2="" or="" π/2="" />< x="" />< π,="" which="" means="" that="" x="" can="" take="" any="" value="" between="" 0="" and="" π,="" excluding="" π/2.="" π,="" which="" means="" that="" x="" can="" take="" any="" value="" between="" 0="" and="" π,="" excluding="" />