Question

If 4 cos⁻¹ x + sin⁻¹ x = π, then the value of x is
(a) $\frac{3}{2}$

(b) $\frac{1}{\sqrt{2}}$

(c) $\frac{\sqrt{3}}{2}$

(d) $\frac{2}{\sqrt{3}}$

Answer 1

(c) $\frac{\sqrt{3}}{2}$
We know that ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}$.

$$\begin{gathered} 4{\cos }^{-1}x+{\sin }^{-1}x=\mathrm{\pi } \\ \Rightarrow 4{\cos }^{-1}x+\frac{\mathrm{\pi }}{2}-{\cos }^{-1}x=\mathrm{\pi } \\ \Rightarrow 3{\cos }^{-1}x=\mathrm{\pi }-\frac{\mathrm{\pi }}{2} \\ \Rightarrow 3{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\cos }^{-1}x=\frac{\mathrm{\pi }}{6} \\ \Rightarrow x=\cos \frac{\mathrm{\pi }}{6} \\ \Rightarrow x=\frac{\sqrt{3}}{2} \end{gathered}$$