Question

Let cos (α + β) = $\frac {4}{5}$ and sin (α - β) = $\frac {5}{13}$, where 0 < α, β < $\frac {π}{4}$ , then tan 2α=?

• A. $\frac {20}{7}$
• B. $\frac {56}{33}$
• C. $\frac {19}{12}$
• D. $\frac {25}{16}$

Answer 1

Answer: (B)

$\frac {56}{33}$

Answer 2

tan2α = $\frac {sin\ 2α}{cos\ 2α}$
tan2α = $\frac {sin \ 2α}{1-2sin^2 \ α}$
sin2α = sin[(α+β)+(α−β)]
sin2α = sin(α+β)cos(α−β) + sin(α−β)cos(α+β)
sin2α = $\frac {3}{5}$ x $\frac {12}{13}$ + $\frac {5}{13}$ x $\frac {4}{5}$
sin2α = $\frac {36}{65}$ + $\frac {20}{65}$
cos2α = 1 - 2 sin2 α
cos2α =1-2 x $(\frac {56}{65})^2$
cos2α =$\frac {33}{56}$​
tan2α = $\frac {(56/65)}{(33/65)}$
tan2α =$\frac {56}{33}$

Therefore, the correct option is (B) $\frac {56}{33}$