Sin^2 (45+a) - sin^2 (30-a) - sin15cos (15+2a) = sin2a

45,30,15 - градусы

Cos^2 (45-a)

cos^2 (60+a) - cos75sin (75-2a) = sin2a

75,60,45 - градусы

1) sin^2 (45+a) - sin^2 (30-a) - sin 15*cos (15+2a) =

(sin 45*cos a + cos 45*sin a) ^2 - (sin 30*cos a - cos 30*sin a) ^2 -

- sin 15 * (cos 15*cos 2a - sin 15*sin 2a) =

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

- sin 15*cos 15*cos 2a + sin^2 15*sin 2a =

= 1/2*cos^2 a + 2*1/2*cos a*sin a + 1/2*sin^2 a - 1/4*cos^2 a +

+ 2*√3/4*sin a*cos a - 3/4*sin^2 a - 1/2*sin 30*cos 2a + (1-cos 30) / 2*sin 2a =

= cos^2 a * (1/2 - 1/4) + sin^2 a * (1/2 - 3/4) + sin 2a * (1/2 + √3/4 + 1/2 - √3/4) -

- 1/4*cos 2a =

= 1/4*cos^2 a - 1/4*sin^2 a + sin 2a * (1 + 0) - 1/4*cos 2a = sin 2a

2) доказывается точно также

cos^2 (45-a) + cos^2 (60+a) - cos 75*sin (75-2a) =

= (cos 45*cos a + sin 45*sin a) ^2 + (cos 60*cos a - sin 60*sin a) ^2 -

- cos (90-15) * sin (90-15-2a) =

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

- sin 15*cos (15+2a) =

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

- sin 15 * (cos 15*cos 2a - sin 15*sin 2a)

Этот пример абсолютно совпадает с 1) и тоже равен sin 2a