Prove that : tan^-1x + tan^-1y + tan^-1z = pie/2

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Solution

$\tan^{1} x + \tan^{- 1} y + \tan^{- 1} z = π / 2 So \tan^{1} x + \tan^{- 1} y = π / 2 - \tan^{- 1} z As \tan^{- 1} x + \cot^{- 1} x = \frac{π}{2}, so \tan^{1} x + \tan^{- 1} y = \cot^{- 1} z As \cot^{- 1} x = \tan^{- 1} \frac{1}{x} So \tan^{1} x + \tan^{- 1} y = \tan^{- 1} \frac{1}{z} \tan^{- 1} (\frac{x + y}{1 - xy}) = \tan^{- 1} \frac{1}{z} So \frac{x + y}{1 - xy} = \frac{1}{z} zx + zy = 1 - xy So zx + zy + xy = 1$

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