If x^2 +y^2 +z^2 = r^2 , then tan^(−1) (((xy)/(zr)))+tan^(−1) (((yz)/(xr)))+tan^(−1) (((xz)/(yr))) =


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Question Number 30939 by paddu1234 last updated on 01/Mar/18

$If x^{2} + y^{2} + z^{2} = r^{2}, then$ $\tan^{- 1} (\frac{x y}{z r}) + \tan^{- 1} (\frac{y z}{x r}) + \tan^{- 1} (\frac{x z}{y r}) =$
	Answered by ajfour last updated on 01/Mar/18

	$l e t \tan θ = \frac{x y}{z r}, \tan ϕ = \frac{y z}{x r},$ $a n d \tan ψ = \frac{x z}{y r}$ $\Rightarrow \tan (θ + ϕ + ψ) = \frac{\frac{x y}{z r} + \frac{y z}{x r} + \frac{z x}{y r} - \frac{x y z}{r^{3}}}{1 - (\frac{y^{2}}{r^{2}} + \frac{z^{2}}{r^{2}} + \frac{x^{2}}{r^{2}})}$ $\Rightarrow \tan (θ + ϕ + ψ) = n o t d e f i n e d$ $h e n c e θ + ϕ + ψ = \pm \frac{π}{2}, o r m a y b e$ $= \pm \frac{3 π}{2} .$
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