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

Options

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

Answer

To solve the problem \( \cos\left(\frac{1}{2} \cos^{-1}\left(\frac{1}{8}\right)\right) \), we will follow these steps:

### Step 1: 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 \( \cos(\theta) \)

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 expression inside the square root:
\[ 1 + \frac{1}{8} = \frac{8}{8} + \frac{1}{8} = \frac{9}{8} \]
Now, substitute this back into the 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:
\[ \frac{3}{4} \]

### Final Answer:
\[ \frac{3}{4} \]