no. of ordered pair

[man111]

(1) Find No. of ordered pair satisfying $\displaystyle\frac{sin^{-1}x}{x}+\frac{sin^{-1}y}{y}=2$

(2) No. of solution of the equation $(\pi )cot^{-1}(x-1)+(\pi-1) cot^{-1}(x) = 2\pi-1$

[Руст]

1) Let $f(x)=\frac{\sin^{-1}x}{x}$. It defined for $x\in [-1,1]$ (we can define $f(0)=1$) and $f(-x)=f(x)$. Because $f(x)>1$ if $x\not =0$ $f(x)+f(y)>2$ if $(x,y)\not =(0,0)$.
Unique solution $(x,y)=(0,0)$.

2) Let $f(x)=\pi cot^{-1}(x-1)+(\pi -1)cot^{-1}(x)$. This function is continiosly and $f'(x)=\frac{-\pi}{(x-1)^2+1}+\frac{1-\pi}{x^2-1}<0$. $f(-\infty)=\pi(2\pi-1)>2\pi -1$ and $f(\infty)=0<2\pi -1$. Therefore equation had unique solution.

[man111]

Thanks pyctr.