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Find the value of $\mathrm{ arctan\ \frac{1}{2}+arctan\ \frac{1}{3}.}$
Option: 1
$\frac{\pi}{3}$

Option: 2
$\frac{\pi}{4}$

Option: 3
$\frac{\pi}{2}$

Option: 4
$\frac{\pi}{6}$

Answers (1)

best_answer

option (b) $\frac{\pi}{4}$

Given that,

$\mathrm{ arctan\ \frac{1}{2}+arctan\ \frac{1}{3} }$

Cosider,

$\mathrm{A=arctan\ \frac{1}{2} }$

$\mathrm{tan\ A=\frac{1}{2}}$

$\mathrm{B=arctan\ \frac{1}{3} }$

$\mathrm{tan\ B=\frac{1}{3} }$

Thus,

$\mathrm{ A+B=\theta }$

$\mathrm{ tan\left (A+B \right )=tan\ \theta }$

$\mathrm{\frac{tan\ A+tan\ B}{1-tan\ A\ tan\ B}=tan\ \theta }$

$\mathrm{\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}\times \frac{1}{3}}=tan\ \theta }$

$\mathrm{tan\ \theta=1 }$

$\mathrm{\theta=\frac{\pi}{4} }$

$\mathrm{arctan\ \frac{1}{2}+arctan\ \frac{1}{3}=\frac{\pi}{4} }$

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shivangi.shekhar

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