Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

$0\le \alpha ,\beta \le \frac{\pi }{4}; \alpha +\beta \le \frac{\pi }{2}$

$\cos (\alpha +\beta )=4/5\Rightarrow \tan (\alpha +\beta )=3/4$

$\sin (\alpha -\beta )=5/13\Rightarrow \alpha -\beta \in {\theta }_1$

and $\tan (\alpha -\beta )=5/12$

$\tan 2\alpha =\tan (\alpha +\beta +\alpha -\beta )$

$=\frac{\tan (\alpha +\beta )+\tan (\alpha -\beta )}{1-\tan (\alpha +\beta )\tan (\alpha -\beta )}$

$=\frac{3/4+5/12}{1-3/4\times \frac{5}{12}}=\frac{56}{33}$