Question

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

Answer 1

We know that ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$

$$\begin{gathered} \therefore 4{\sin }^{-1}x+{\cos }^{-1}x=\mathrm{\pi } \\ \Rightarrow 4{\sin }^{-1}x+\frac{\mathrm{\pi }}{2}-{\sin }^{-1}x=\pi \left[\because {\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}\right] \\ \Rightarrow 3{\sin }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\sin }^{-1}x=\frac{\mathrm{\pi }}{6} \\ \Rightarrow x=\sin \frac{\mathrm{\pi }}{6} \\ \Rightarrow x=\frac{1}{2} \end{gathered}$$

∴ $x=\frac{1}{2}$