The two terms on the left side both have a common factor of (sin(α) + cos(β)), so we can pull it out like so:
(sin(α) + cos(β))² + (cos(β) + sin(α)) (cos(β) - sin(α))
= (sin(α) + cos(β)) [(sin(α) + cos(β)) + (cos(β) - sin(α))]
= (sin(α) + cos(β)) (sin(α) + cos(β) + cos(β) - sin(α))
= 2 cos(β) (sin(α) + cos(β))
