Question

If $\cos (\alpha+\beta)=\frac{4}{5}, \sin (\alpha-\beta)=\frac{5}{13}$ and $\alpha, \beta$ between 0 and $\frac{\pi}{4}$, then $\tan 2 \alpha$ is equal to

(A) $\frac{56}{33}$

(B) $\frac{33}{56}$

(C) $\frac{16}{65}$

(D) $\frac{60}{61}$

Answer 1

The correct option will be (A).

HINT —

$\because \cos (\alpha+\beta)=\frac{4}{5}$

$\begin{array}{cl}\Rightarrow & \tan (\alpha+\beta)=\frac{3}{4} \text { and } \sin (\alpha-\beta)=\frac{5}{13} \\\Rightarrow & \tan (\alpha-\beta)=\frac{5}{12}\end{array}$

Now, $\tan 2 \alpha=\tan [(\alpha+\beta)+(\alpha-\beta)]$