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4. tan^-1 1/5 - tan^-1 1/70 + tan^-1 1/99 = pie/4

$4 \tan^{- 1} (\frac{1}{5}) = 2 \times 2 \tan^{- 1} (\frac{1}{5}) = 2 [\tan^{- 1} \frac{2 / 5}{1 - (1 / 25)}] (u \sin g, 2 \tan^{- 1} A = \frac{2 \tan A}{1 - \tan^{2} A}) = 2 [\tan^{- 1} (\frac{2 / 5}{24 / 25})] = 2 \tan^{- 1} (\frac{5}{12}) = \tan^{- 1} \frac{2 (5 / 12)}{1 - (25 / 144)} = \tan^{- 1} \frac{10 / 12}{119 / 144} = \tan^{- 1} \frac{120}{119} \Rightarrow 4 \tan^{- 1} (\frac{1}{5}) = \tan^{- 1} \frac{120}{119} N o w, 4 \tan^{- 1} (\frac{1}{5}) - \tan^{- 1} (\frac{1}{70}) + \tan^{- 1} (\frac{1}{99}) = \tan^{- 1} \frac{120}{119} - [\tan^{- 1} (\frac{1}{70}) - \tan^{- 1} (\frac{1}{99})] = \tan^{- 1} \frac{120}{119} - \tan^{- 1} \frac{\frac{1}{70} - \frac{1}{99}}{1 + (\frac{1}{99}) (\frac{1}{77})} (u \sin g \tan^{- 1} A - \tan^{- 1} B = \tan^{- 1} (\frac{A - B}{1 + A B})) = \tan^{- 1} \frac{120}{119} - \tan^{- 1} \frac{29}{6931} = \tan^{- 1} \frac{120}{119} - \tan^{- 1} \frac{1}{239} = \tan^{- 1} \frac{\frac{120}{119} - \frac{1}{239}}{1 + (\frac{120}{119}) (\frac{1}{239})} = \tan^{- 1} \frac{\frac{120}{119} - \frac{1}{239}}{1 + (\frac{120}{119}) (\frac{1}{239})} = \tan^{- 1} \frac{120 \times 239 - 119}{119 \times 239 + 120} = \tan^{- 1} 1 = \tan^{- 1} (\tan \frac{π}{4}) = \frac{π}{4} = R . H . S . H e n c e p r o v e d .$

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