^ -1 3+ ^ -1 = ^ -1 (3+ 1-3 ) is valid for what values of ? - Vidyadip

Question

$\tan ^{-1} 3+\tan ^{-1} \lambda=\tan ^{-1}\left(\frac{3+\lambda}{1-3 \lambda}\right)$ is valid for what values of $\lambda$ ?

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Answer

$\text {Given, } \tan ^{-1} 3+\tan ^{-1} \lambda=\tan ^{-1}\left(\frac{3+\lambda}{1-3 \lambda}\right)$
$\tan ^{-1} 3+\tan ^{-1} \lambda=\tan ^{-1}\left(\frac{3+\lambda}{1-3 \lambda}\right) \text { for } 3 \lambda<1$
$\therefore 3 \lambda<1$
$\Rightarrow \lambda<\frac{1}{3}$

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