Question

If $\alpha$ and $\beta$ lies between 0 and $\frac{\pi}{2}$ and if $\cos (\alpha+\beta)=\frac{12}{13}$ and $\sin (\alpha-\beta)=\frac{3}{5}$, then value of $\sin 2 \alpha$ is

• A. $\frac{55}{56}$
• B. $\frac{13}{58}$
• C. $0$
• D. $\frac{56}{65}$

Answer 1

Answer: (D)

$\sin (2 \alpha)=\sin (\alpha+\beta+\alpha-\beta)$
$=\sin (\alpha+\beta) \cos (\alpha-\beta)+\cos (\alpha+\beta) \sin (\alpha-\beta)$
$=\frac{5}{13} \cdot \frac{4}{5}+\frac{12}{13} \cdot \frac{3}{5}=\frac{56}{65}$