Question

Write the value of $\tan \left(2 \tan ^{-1} \frac{1}{5}\right)$

Answer 1

SOLUTION : Since $2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^2}$ for $|x|<1$

$\text { so }, 2 \tan ^{-1}\left(\frac{1}{5}\right)=\tan ^{-1}\left(\frac{2 \times \frac{1}{5}}{1-\left(\frac{1}{5}\right)^2}\right)=\tan ^{-1}\left(\frac{2}{\frac{5}{24}}\right)=\tan ^{-1}\left(\frac{5}{12}\right) \quad \therefore \quad \tan \left(\tan ^{-1} \frac{5}{12}\right)=\frac{5}{12}$