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Question
If tan−1 3 + tan−1 x = tan−1 8, then x =
Options
5
1/5
5/14
14/5
Solution
(b) `1/5`
We know that
\[\tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} \frac{x + y}{1 - xy}\]
Now,
\[\tan^{- 1} 3 + \tan^{- 1} x = \tan^{- 1} 8\]
\[ \Rightarrow \tan^{- 1} \left( \frac{3 + x}{1 - 3x} \right) = \tan^{- 1} 8\]
\[ \Rightarrow \frac{3 + x}{1 - 3x} = 8\]
\[ \Rightarrow 3 + x = 8 - 24x\]
\[ \Rightarrow 3 - 8 = - 24x - x\]
\[ \Rightarrow - 5 = - 25x\]
\[ \Rightarrow x = \frac{5}{25} = \frac{1}{5}\]
\[\]
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