Solveeit Logo
Login

Question

Mathematics Question on Inverse Trigonometric Functions

The value of cos⁡(12cos⁡−118)\cos \left(\frac{1}{2} \cos^{-1} \frac{1}{8}\right) is equal to

A

34\frac{3}{4}

B

−34 - \frac{3}{4}

C

116\frac{1}{16}

D

14\frac{1}{4}

Answer

34\frac{3}{4}

Explanation

Solution

Let cos⁡−118=θ where 0<θ<π\cos^{-1} \frac{1}{8} = \theta \,where \,0 < \theta < \pi ⇒cos⁡(12cos⁡−118)=cos⁡θ2\Rightarrow \cos\left(\frac{1}{2} \cos^{-1} \frac{1}{8}\right) = \cos \frac{\theta}{2} Now cos⁡−118=θ⇒cos⁡θ=18 \cos^{-1} \frac{1}{8 } = \theta \Rightarrow \cos\theta = \frac{1}{8} ⇒2cos⁡2θ2−1=18⇒cos⁡θ2=34\Rightarrow 2 \cos^{2} \frac{\theta}{2} - 1 = \frac{1}{8} \Rightarrow \cos \frac{\theta}{2} = \frac{3}{4} [∵0<θ2<π2,cos⁡π2≠−34]\left[\because 0 < \frac{\theta}{2} < \frac{\pi}{2}, \cos \frac{\pi}{2} \ne - \frac{3}{4}\right]