Question

If 4 cos⁻¹ x + sin⁻¹ x = π, then the value of x is

 

Options

• `2/3`

• `1/sqrt2`

• `sqrt3/2`

• `2/sqrt3`

Answer 1

(c) `sqrt3/2`

We know that 
\[\sin^{- 1} x + \cos^{- 1} x = \frac{\pi}{2}\]
\[4 \cos^{- 1} x + \sin^{- 1} x = \pi\]
\[ \Rightarrow 4 \cos^{- 1} x + \frac{\pi}{2} - \cos^{- 1} x = \pi\]
\[ \Rightarrow 3 \cos^{- 1} x = \pi - \frac{\pi}{2}\]
\[ \Rightarrow 3 \cos^{- 1} x = \frac{\pi}{2}\]
\[ \Rightarrow \cos^{- 1} x = \frac{\pi}{6}\]
\[ \Rightarrow x = \cos\frac{\pi}{6}\]
\[ \Rightarrow x = \frac{\sqrt{3}}{2}\]