Question

The value of \(\cos \left( \frac12 \cos^-1 \frac18 \right)\) is:

• A. \(\frac34\)
• B. \(-\frac34\)
• C. \(\frac116\)
• D. \(\frac14\)

Hint:

For half-angle problems involving inverse cosine, always ensure the sign of the result matches the quadrant of the half-angle.

Answer 1

The Correct Option is A

Let \(\theta = \cos^-1 \frac18\). Then \(\cos \theta = \frac18\).
We need \(\cos \frac\theta2\). Recall the half-angle formula:
\[ \cos \frac\theta2 = \sqrt \frac1 + \cos \theta2 \] Since \(0 \leq \theta \leq \pi\), \(\cos \frac\theta2>0\), so we take the positive root.
Substitute \(\cos \theta = \frac18\):
\[ \cos \frac\theta2 = \sqrt \frac1 + \frac182 \] Simplify:
\[ 1 + \frac18 = \frac98 \] So:
\[ \cos \frac\theta2 = \sqrt \frac\frac982 = \sqrt \frac916 = \frac34 \] Thus, the value is \(\frac34\).