Which of the following is/are the value of `cos [(1)/(2) cos^(-1) (cos (-(14pi)/(5)))]` ?

Options

A. `cos(-(7pi)/(5))`
B. `sin((pi)/(10))`
C. `cos((2pi)/(5))`
D. `-cos((3pi)/(5))`

Explanation

To find the value of \( \cos \left( \frac{1}{2} \cos^{-1} \left( \cos \left( -\frac{14\pi}{5} \right) \right) \right) \), we will follow these steps:

### Step 1: Simplify \( \cos \left( -\frac{14\pi}{5} \right) \)
Using the property of cosine that \( \cos(-x) = \cos(x) \), we can rewrite:
\[ \cos \left( -\frac{14\pi}{5} \right) = \cos \left( \frac{14\pi}{5} \right) \]

### Step 2: Reduce \( \frac{14\pi}{5} \) to a standard angle
To simplify \( \frac{14\pi}{5} \), we can subtract \( 2\pi \) (which is \( \frac{10\pi}{5} \)):
\[ \frac{14\pi}{5} - 2\pi = \frac{14\pi}{5} - \frac{10\pi}{5} = \frac{4\pi}{5} \]
Thus, we have:
\[ \cos \left( \frac{14\pi}{5} \right) = \cos \left( \frac{4\pi}{5} \right) \]

### Step 3: Substitute back into the expression
Now we substitute this back into the original expression:
\[ \cos \left( \frac{1}{2} \cos^{-1} \left( \cos \left( \frac{4\pi}{5} \right) \right) \right) \]

### Step 4: Use the property of inverse cosine
Using the property \( \cos^{-1}(\cos(x)) = x \) for \( x \) in the range of \( [0, \pi] \), we find:
\[ \cos^{-1} \left( \cos \left( \frac{4\pi}{5} \right) \right) = \frac{4\pi}{5} \]
since \( \frac{4\pi}{5} \) is within the range of \( [0, \pi] \).

### Step 5: Calculate \( \frac{1}{2} \frac{4\pi}{5} \)
Now we compute:
\[ \frac{1}{2} \cdot \frac{4\pi}{5} = \frac{2\pi}{5} \]

### Step 6: Find \( \cos \left( \frac{2\pi}{5} \right) \)
Finally, we need to find:
\[ \cos \left( \frac{2\pi}{5} \right) \]

### Final Result
Thus, the value of \( \cos \left( \frac{1}{2} \cos^{-1} \left( \cos \left( -\frac{14\pi}{5} \right) \right) \right) \) is:
\[ \cos \left( \frac{2\pi}{5} \right) \]