Question

The value of $\cos \left(\frac{1}{2} \cos^{-1} \frac{1}{8}\right) $ is equal to

• A. $\frac{3}{4}$
• B. $ - \frac{3}{4}$
• C. $\frac{1}{16}$
• D. $\frac{1}{4}$

Answer 1

Answer: (A)

Let $\cos^{-1} \frac{1}{8} = \theta \,where \,0 < \theta < \pi $
$ \Rightarrow \cos\left(\frac{1}{2} \cos^{-1} \frac{1}{8}\right) = \cos \frac{\theta}{2} $
Now $ \cos^{-1} \frac{1}{8 } = \theta \Rightarrow \cos\theta = \frac{1}{8}$
$ \Rightarrow 2 \cos^{2} \frac{\theta}{2} - 1 = \frac{1}{8} \Rightarrow \cos \frac{\theta}{2} = \frac{3}{4} $
$ \left[\because 0 < \frac{\theta}{2} < \frac{\pi}{2}, \cos \frac{\pi}{2} \ne - \frac{3}{4}\right] $