First slide

Properties of ITF

Question

$4 \tan^{- 1} \frac{1}{5} - \tan^{- 1} \frac{1}{239}$ is equal to

Moderate

A

$π$
B

$π / 2$
C

$π / 3$
D

$π / 4$

Solution

We have,
$4 \tan^{- 1} \frac{1}{5} - \tan^{- 1} \frac{1}{239}$
$\begin{array}{l} = 2 \tan^{- 1} (\frac{2 / 5}{1 - 1 / 25}) - \tan^{- 1} \frac{1}{239} \\ = 2 \tan^{- 1} (5 / 12) - \tan^{- 1} 1 / 239 \\ = \tan^{- 1} (\frac{2 (5 / 12)}{1 - (5 / 12)^{2}}) - \tan^{- 1} \frac{1}{239} \end{array}$
$\begin{array}{l} = \tan^{- 1} \frac{120}{119} - \tan^{- 1} \frac{1}{239} \\ = \tan^{- 1} (\frac{120 / 119 - 1 / 239}{1 + 120 / 119 \times 1 / 239}) \\ = \tan^{- 1} (\frac{28569}{28569}) = \tan^{- 1} (1) = \frac{π}{4} \end{array}$

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