Let α and β be real numbers such that - π 4 < β < 0 < α < π 4 . If sin α + β = 1 3 and cos α - β = 2 3 , then the greatest integer less than or equal to sin α cos β + cos β sin α + cos α sin β + sin β cos α 2 is

Step-by-step solution

sin α cos β + cos α sin β + cos β sin α + sin β cos α 2 = cos α - β sin β cos β + cos α - β sin α cos α 2 = 2 cos α - β 1 sin 2 β + 1 sin 2 α 2 = 16 9 2 sin α + β · cos α - β sin 2 α · sin 2 β 2 = 16 9 4 · 1 3 · 2 3 cos 2 α - 2 β - cos 2 α + 2 β 2 = 16 9 8 9 2 cos 2 α - β - 1 - 1 - 2 sin 2 α + β 2 = 16 9 8 9 8 9 - 2 + 2 9 2 = 16