Question

If $\tan (\alpha +\beta )=1/3$ & $\tan (\alpha -\beta )=1/4$. What is the value of $\tan 2\alpha$?

Answer 1

We have,

$\tan \left|\alpha +\beta \right|=\frac{1}{3}$ $\tan \left|\alpha -\beta \right|=\frac{1}{4}$

$\alpha +\beta ={\tan }^{-1}\frac{1}{3}$ $\alpha -\beta ={\tan }^1\frac{1}{4}$

$2\alpha ={\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{1}{4}$

$={\tan }^{-1}\left(\frac{\frac{1}{3}+\frac{1}{4}}{1-\frac{1}{12}}\right)$

$\tan 2\alpha =\frac{7}{11}$

We get, $\tan 2\alpha =\frac{7}{11}.$

Hence, this is the answer.