Question

If ${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z+{\sin }^{-1}t=2\mathrm{\pi }$, then find the value of x² + y² + z² + t²

Answer 1

We know that the maximum value of ${\sin }^{-1}x,{\sin }^{-1}y,{\sin }^{-1}z\mathrm{and}{\sin }^{-1}t\mathrm{is}\frac{\mathrm{\pi }}{2}$
Now,
$$\begin{gathered} \mathrm{LHS}={\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z+{\sin }^{-1}t \\ =\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2}+\frac{\mathrm{\pi }}{2} \\ =2\mathrm{\pi }=\mathrm{RHS} \end{gathered}$$
Now,
$$\begin{gathered} {\sin }^{-1}x=\frac{\mathrm{\pi }}{2},{\sin }^{-1}y=\frac{\mathrm{\pi }}{2},{\sin }^{-1}z=\frac{\mathrm{\pi }}{2}\mathrm{and}{\sin }^{-1}t=\frac{\mathrm{\pi }}{2} \\ \Rightarrow x=\sin \frac{\mathrm{\pi }}{2},y=\sin \frac{\mathrm{\pi }}{2},z=\sin \frac{\mathrm{\pi }}{2}\mathrm{and}t=\sin \frac{\mathrm{\pi }}{2} \\ \Rightarrow x=1,y=1,z=1\mathrm{and}t=1 \\ \therefore x^2+y^2+z^2+t^2=1+1+1+1=4 \end{gathered}$$