If α and β be between 0 and π/2 and if cos(α + β) = 12/13 and sin(α – β) = 3/5 then sin 2α is: (a) 16/15

Question

If α and β be between 0 and \(\frac{\pi}{2}\) and if cos(α + β) = \(\frac{12}{13}\) and sin(α – β) = \(\frac{3}{5}\) then sin 2α is: 

(a) \(\frac{16}{15}\)

(b) 0 

(c) \(\frac{56}{65}\)

(d) \(\frac{64}{65}\)

Answer 1

(c) \(\frac{56}{65}\)

Given that cos(α + β) = \(\frac{12}{13}\)

∴ sin(α + β) = \(\frac{5}{13}\)

Also given that sin(α – β) = \(\frac{3}{5}\)

∴ cos(α – β) = \(\frac{4}{5}\)

sin 2α = sin[(α + β) + (α – β)] 

= sin(α + β) cos(α – β) + cos(α + β) sin(α – β)