Find the value of
Question:

Find the value of 4 tan-1 1/5 – tan-1 1/239

Solution:

4 tan-1 1/5 – tan-1 1/239

= 2 (tan-1 1/5) – tan-1 1/239

$=2 \tan ^{-1} \frac{\frac{2}{5}}{1-\left(\frac{1}{5}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=2 \tan ^{-1} \frac{2 / 5}{24 / 25}-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{5}{12}-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{\frac{2}{5}}{1-\left(\frac{1}{5}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=2 \tan ^{-1} \frac{2 / 5}{24 / 25}-\tan ^{-1} \frac{1}{239}$

= 2 tan-1 5/12 – tan-1 1/239

$=\tan ^{-1} \frac{2 \cdot \frac{5}{12}}{1-\left(\frac{5}{12}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=\tan ^{-1} \frac{144 \times 5}{119 \times 6}-\tan ^{-1} \frac{1}{239}$

$=\tan ^{-1} \frac{120}{119}-\tan ^{-1} \frac{1}{239}$

$=\tan ^{-1} \frac{\frac{120}{119}-\frac{1}{239}}{1+\frac{120}{119} \cdot \frac{1}{239}} \quad\left(\because \tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}\right)$

$=\tan ^{-1} \frac{120 \times 239-119}{119 \times 239+120}=\tan ^{-1} \frac{28680-119}{28441+120}$

$=\tan ^{-1} \frac{28561}{28561}=\tan ^{-1} 1=\frac{\pi}{4}$

Thus,

4 tan-1 1/5 – tan-1 1/239 = π/4

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