Sin2a+cos2b (бета) + 7=

Спростити вираз

1) cos (pi/4 + а) cos (pi/4 - а) + (1/2) sin^2 (a) = {произведение косинусов} =

(1/2) (cos (pi/4+a + pi/4 - a) + cos (pi/4+a - (pi/4-a))) + (1/2) sin^2 (a) =

(1/2) (cos (pi/2) + cos (2a) + sin^2 (a)) = {cos (pi/2) = 0, cos (2a) = cos^2 (a) - sin^2 (a) }

= (1/2) (cos^2 (a) - sin^2 (a) + sin^2 (a)) = (1/2) cos^2 (a)

2) cos (a-b) - cos (a+b) = {разность косинусов} =

-2sin (((a-b) + (a+b)) / 2) * sin (((a-b) - (a+b)) / 2) = - 2sin (a) sin (-b) = 2sin (a) sin (b)

3) cos (3a) + sin (a) sin (2a) = cos (3a) + (1/2) (cos (a-2a) - cos (a+2a)) =

cos (3a) + (1/2) cos (a) - (1/2) cos (3a) = (1/2) (cos (3a) + cos (a)) = {сумма косинусов}

= cos ((3a+a) / 2) cos ((3a-a) / 2) = cos (2a) cos (a)

4) cos (2a) - cos (a) cos (3a) = cos (2a) - (1/2) (cos (4a) + cos (2a)) = (1/2) (cos (2a) - cos (4a)) =

(1/2) * (-2) * sin (3a) sin (-2a) = sin (3a) sin (a)