Question

Solve for x: If $\left[{\sin }^{-1}{\cos }^{-1}{\sin }^{-1}{\tan }^{-1}x\right]=1$ where[.] denotes the greatest integer function.

Answer 1

${\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z=\frac{3}{2}$
we know ${\sin }^{-1}x\le \frac{\pi }{2}$
$\Rightarrow {\sin }^{-1}x+{\sin }^{-1}y+{\sin }^{-1}z\le \frac{\pi }{\frac{3}{2}}$
and this holds only when $x=y=z=1$
$x^{2000}+y^{2001}+y^{2002}-\frac{9}{x^{2000}}+y^{2001}+z^{2002}$
$1+1+1-\frac{9}{1+1+1}$ $=3-3=0$