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)

$\cos \left(\frac{1}{2} \cos ^{-1} \frac{1}{8}\right)=x$ (let)
$\Rightarrow \cos ^{-1} \frac{1}{8}=2 \cos ^{-1} x$
$=\cos ^{-1}\left(2 x^{2}-1\right)$
$\Rightarrow \frac{1}{8}=2 x^{2}-1$
$\Rightarrow 2 x^{2}=\frac{1}{8}+1=\frac{9}{8}$
$\Rightarrow x^{2}=\frac{9}{16}$
$\Rightarrow x=\sqrt{\frac{9}{16}}=\frac{3}{4}$
$[\because 0 \le x \le 1$]