Single Choice

The value of $$\cos\Bigg(\dfrac{1}{2}cos^{-1}\dfrac{1}{8}\Bigg)$$ is

A$$\dfrac{3}{4}$$
Correct Answer
B$$\dfrac{-3}{4}$$
C$$\dfrac{1}{16}$$
D$$\dfrac{1}{4}$$

Solution

Let $$cos^{-1}\dfrac{1}{8}=\theta,where\,0<\theta<\pi,\,then\,\dfrac{1}{2}cos^{-1}\dfrac{1}{8}=\dfrac{1}{2}\theta$$

$$\cos\Bigg(\dfrac{1}{2}cos^{-1}\dfrac{1}{8}\Bigg)=\cos\dfrac{\theta}{2}$$

Now $$cos^{-1}\dfrac{1}{8}=\theta\,\,\implies\cos\theta=\dfrac{1}{8}\implies\cos^{2}\dfrac{\theta}{2}=\dfrac{9}{16}\implies\cos\dfrac{\theta}{2}=\dfrac{3}{4}$$

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