Question

If 0$<\alpha ,<\frac{\pi }{4}$such that $\cos (\alpha +\beta )=\frac{4}{5}$and $\sin (\alpha -\beta )=\frac{5}{13}$ then value of $\tan 2\alpha$ is

• A. $\frac{56}{65}$
• B. $\frac{56}{23}$
• C. $\frac{56}{33}$
• D. $\frac{56}{36}$

Answer 1

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

REF.Image.

$0<2<\frac{\pi }{4}$

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

$sin(\alpha -\beta )=\frac{5}{13}$

$tan2\alpha =tan\left[\right(\alpha +\beta )+(\alpha -\beta \left)\right]$

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

$=\frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}.\frac{5}{12}}$

$=\frac{\frac{36+20}{48}}{\frac{48-15}{48}}$

$=\frac{56}{33}$