First slide

Properties of ITF

Question

If $x^{2} + y^{2} + z^{2} = r^{2},$ then $\tan^{- 1} (\frac{x y}{z r}) + \tan^{- 1} (\frac{y z}{x r}) + \tan^{- 1} (\frac{x z}{y r}) =$

Moderate

A

$π$
B

$\frac{π}{2}$
C

0
D

none of these

Solution

We have,
$\tan^{- 1} (\frac{x y}{z r}) + \tan^{- 1} (\frac{y z}{x r}) + \tan^{- 1} (\frac{x z}{y r})$
$= \tan^{- 1} \{\frac{\frac{\begin{matrix} x y \end{matrix}}{z r} + \frac{y z}{x r} + \frac{x z}{y r} - \frac{x y z}{r^{3}}}{1 - (\frac{x^{2} + y^{2} + z^{2}}{r^{2}})}\} = \tan^{- 1} \infty = \frac{π}{2}$

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