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

The Correct Option is B

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) \(\frac {56}{33}\)