The value of `cos^(-1)(cos'(14pi)/(3))` is `"…….."`

Explanation

To solve the problem of finding the value of \( \cos^{-1}(\cos(14\pi/3)) \), we can follow these steps:

### Step 1: Simplify the angle
First, we need to simplify the angle \( 14\pi/3 \). Since the cosine function is periodic with a period of \( 2\pi \), we can reduce \( 14\pi/3 \) by subtracting \( 2\pi \) (or \( 6\pi/3 \)):

\[ 14\pi/3 - 6\pi/3 = 8\pi/3 \]

Now, we can simplify \( 8\pi/3 \) further by subtracting another \( 2\pi \):

\[ 8\pi/3 - 6\pi/3 = 2\pi/3 \]

### Step 2: Apply the cosine function
Now we can find the cosine of the simplified angle:

\[ \cos(14\pi/3) = \cos(2\pi/3) \]

### Step 3: Find the value of \( \cos(2\pi/3) \)
The angle \( 2\pi/3 \) is in the second quadrant, where cosine is negative. The reference angle for \( 2\pi/3 \) is \( \pi/3 \), so:

\[ \cos(2\pi/3) = -\cos(\pi/3) = -\frac{1}{2} \]

### Step 4: Use the inverse cosine function
Now we can find the value of \( \cos^{-1}(\cos(14\pi/3)) \):

\[ \cos^{-1}(\cos(14\pi/3)) = \cos^{-1}(\cos(2\pi/3)) \]

### Step 5: Determine the principal value
The principal value of \( \cos^{-1}(x) \) lies in the range \( [0, \pi] \). Since \( 2\pi/3 \) is within this range, we have:

\[ \cos^{-1}(\cos(2\pi/3)) = 2\pi/3 \]

### Final Answer
Thus, the value of \( \cos^{-1}(\cos(14\pi/3)) \) is:

\[ \boxed{\frac{2\pi}{3}} \]