If ` tan ^(-1 )""(yz )/( xr ) + tan ^(-1 )""(zx)/(yr) + tan ^(-1) ""(xy)/(zr) =(pi)/(2) `, prove that , ` x^(2) + y^(2) + z^(2) + r^(2) `

Explanation

Given

`tan ^{-1}((y z) / (x r))+tan ^{-1}((x z) / (y r))+tan ^{-1}((x y) /((z r))=pi / 2`

`=tan ^{-1}(frac{frac{y z}{x r}+frac{z x}{y r}+frac{x y}{z r}-frac{x y z}{r^{3}}}{1-(frac{z^{2}}{r^{2}}+frac{x^{2}}{r^{2}}+frac{y^{2}}{r^{2}})})`

`=tan ^{-1}(frac{frac{y z}{x r}+frac{z x}{y r}+frac{x y}{z r}-frac{x y z}{r^{3}}}{1-(frac{x^{2}+y^{2}+z^{2}}{r^{2}})})`

`=tan ^{-1}(frac{frac{y z}{x r}+frac{z x}{y r}+frac{x y}{z r}-frac{x y z}{r^{3}}}{1-1})`

`=tan ^{-1}(infty)=frac{pi}{2}`