The value of `cos(1/2cos^(-1)(1/8))` is

Options

A. `3/4`
B. `-3/4`
C. `1/16`
D. `1/4`

Explanation

To find the value of \( \cos\left(\frac{1}{2} \cos^{-1}\left(\frac{1}{8}\right)\right) \), we can follow these steps:

### Step 1: Set up the equation
Let \( \theta = \cos^{-1}\left(\frac{1}{8}\right) \). This means that \( \cos(\theta) = \frac{1}{8} \).

### Step 2: Use the half-angle formula
We need to find \( \cos\left(\frac{\theta}{2}\right) \). The half-angle formula for cosine is given by:
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \]

### Step 3: Substitute the value of \( \cos(\theta) \)
Substituting \( \cos(\theta) = \frac{1}{8} \) into the half-angle formula:
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{1}{8}}{2}} \]

### Step 4: Simplify the expression
First, simplify the numerator:
\[ 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \]
Now substitute this back into the half-angle formula:
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{\frac{9}{8}}{2}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \]

### Step 5: Conclusion
Thus, the value of \( \cos\left(\frac{1}{2} \cos^{-1}\left(\frac{1}{8}\right)\right) \) is:
\[ \frac{3}{4} \]