Q59 / 120 · VITEEE 2016 · Mathematics
The value of $\cos \left(\frac{1}{2} \cos ^{-1} \frac{1}{8}\right)$ is equal to
- A$-3 / 4$
- Correct answerB$3 / 4$
- C$1 / 16$
- D$1 / 4$
Solution
Let $\cos ^{-1} \frac{1}{8}=\theta$, where $0 < \theta < \frac{\pi}{2}$
$\begin{array}{l}
\Rightarrow \frac{1}{2} \cos ^{-1} \frac{1}{8}=\frac{1}{2} \theta \\
\Rightarrow \cos \left(\frac{1}{2} \cos ^{-1} \frac{1}{8}\right)=\cos \frac{1}{2} \theta \\
\text { Now, } \cos ^{-1} \frac{1}{8}=\theta \Rightarrow \cos \theta=\frac{1}{8} \\
\Rightarrow 2 \cos ^{2} \frac{\theta}{2}-1=\frac{1}{8}
\end{array}$
$\begin{aligned} \Rightarrow \cos ^{2} \frac{\theta}{2}=& \frac{9}{16} \Rightarrow \cos \frac{\theta}{2}=\frac{3}{4} \\ &\left[\because 0 < \frac{\theta}{2} < \frac{\pi}{4}, \text { so } \cos \frac{\theta}{2} \neq-\frac{3}{4}\right] \end{aligned}$
Hence, the correct answer is option: 'B'.