Question

Solve: ${\cot }^{-1}\left(\frac{xy+1}{x-y}\right)+{\cot }^{-1}\left(\frac{yz+1}{y-z}\right)+{\cot }^{-1}\left(\frac{zx+1}{z-x}\right)$

Answer 1

$co{t}^{-1}\left(\frac{xy+1}{x-y}\right)+co{t}^{-1}\left(\frac{yz+1}{y-z}\right)+co{t}^{-1}\left(\frac{xz+1}{z-x}\right)$

$=ta{n}^{-1}\left(\frac{x-y}{1+xy}\right)+ta{n}^{-1}\left(\frac{y-z}{1+yz}\right)+ta{n}^{-1}\left(\frac{z-x}{1+xz}\right)$

$=ta{n}^{-1}x-ta{n}^{-1}y+ta{n}^{-1}y-ta{n}^{-1}y-ta{n}^{-1}z+ta{n}^{-1}z-ta{n}^{-1}x$

=0