Question

Prove: (sin α + cos β)^2 + (cos β + sin α)(cos β − sin α) = 2 cos β(sin α + cos β)

Answer 1

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))