$\tan ^{-1} \frac{1}{11}+\tan ^{-1} \frac{2}{11}$ is equal to

Question.
$\tan ^{-1} \frac{1}{11}+\tan ^{-1} \frac{2}{11}$ is equal to
(a) 0
(b) $1 / 2$
(c) $-1$
(d) none of these

Solution:
(d) none of these
We know that $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$.
Now,
$\tan ^{-1} \frac{1}{11}+\tan ^{-1} \frac{2}{11}=\tan ^{-1}\left(\frac{\frac{1}{11}+\frac{2}{11}}{1-\frac{1} {11} \frac{2}{11}}\right)$
$=\tan ^{-1}\left(\frac{\frac{3}{11}}{\frac{121-2}{121}}\right)$
$=\tan ^{-1}\left(\frac{\frac{3}{11}}{\frac{119}{121}}\right)$
$=\tan ^{-1}\left(\frac{33}{119}\right)$
$=0.27$

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