Question

If $\cos (\alpha +\beta )=\frac{4}{5}$ and $\sin (\alpha -\beta )=\frac{5}{13}$ where $\alpha ,\beta$ lie between $0^{\circ }$ and ${45}^{\circ }$, then the value of $\tan 2\alpha$ is

• A. $\frac{56}{33}$
• B. $\frac{46}{33}$
• C. $\frac{14}{33}$
• D. $\frac{16}{33}$

Answer 1

The correct option is A $\frac{56}{33}$

$\cos (\alpha +\beta )=\frac{4}{5},\sin (\alpha -\beta )=\frac{5}{13}\sin (\alpha +\beta )=\sqrt{1-{\left(\frac{4}{5}\right)}^2}=\frac{3}{5}\cos (\alpha -\beta )=\frac{12}{13}\sin (\alpha +\beta +\alpha -\beta )=\sin (\alpha +\beta )\cos (\alpha -\beta )+\sin (\alpha -\beta )\cos (\alpha +\beta )=\frac{3}{5}\times \frac{12}{13}+\frac{5\times 4}{5\times 13}=\frac{56}{65}\cos (\alpha +\beta +\alpha -\beta )=\cos (\alpha +\beta )\cos (\alpha -\beta )-\sin (\alpha +\beta )\sin (\alpha -\beta )=\frac{4}{5}\times \frac{12}{13}-\frac{5}{13}\times \frac{3}{5}=\frac{33}{65}\therefore \tan 2\alpha =\frac{\sin 2\alpha }{\cos 2\alpha }=\frac{56\times 65}{65\times 33}=\frac{56}{33}$