First slide

Properties of ITF

Question

If $\cot^{- 1} x + \cot^{- 1} y + \cot^{- 1} z = \frac{π}{2},$ then $x + y + z$ is also equal to

Moderate

A

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$
B

$x y + y z + z x$
C

$x y z$
D

None of these

Solution

$\cot^{- 1} x + \cot^{- 1} y + \cot^{- 1} z = \frac{π}{2},$
$\begin{array}{l} \Rightarrow \frac{π}{2} - \tan^{- 1} x + \frac{π}{2} - \tan^{- 1} y + \frac{π}{2} - \tan^{- 1} z = \frac{π}{2} \\ \Rightarrow \tan (\tan^{- 1} x + \tan^{- 1} y + \tan^{- 1} z) = \tan π \\ \Rightarrow \tan (\frac{x + y + z - x y z}{1 - (x y + y z + z x)}) = 0 \\ \Rightarrow x + y + z = x y z \end{array}$

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