1) sin 82° 30' cos 37° 30'

sina+sinb=2sin(a+b/2)*cos(a-b/2)

a+b/2=82,5

a-b/2=37,5

a+b=165

a-b=75

a=165-b

165-b-b=75

-2b=-90

b=45

a=120

(1/2)* 2sin 82° 30' cos 37° 30'=(sin120+sin45)/2=(sin(180-60)+sin45)/2=(sin60+sin45)/2=(sqrt(3)/2+sqrt(2)/2)/2=((sqrt(3)+sqrt(2))/2)/2=(sqrt(3)+sqrt(2))/4

 2) 2sin 3a cos 4a + sin a = sin 7a

 sina1+sinb=2sin(a1+b/2)*cos(a1-b/2)

a1+b/2=3a

a1-b/2=4a

a1+b=6a

a1-b=8a

a1=6a-b

6a-b-b=8a

-2b=2a

b=-a

a1=6a+a=7a

2sin 3a cos 4a + sin a = sin 7a

sin7a+sin(-a)+sina=sin7a

sin7a-sina+sina=sin7a

sin7a=sin7a

3) 4cos a/2 cos a sin 3a/2 = sin a + sin 2a + sin 3a

 2*2cosa cosa/2 sin3a/2=sina + sin2a + sin3a

2*(cosa/2 + cos 3a/2)*sin3a/2=sina + sin2a + sin3a

2 cosa/2 * sin3a/2  + 2 cos3a/2 sin3a/2 = sina + sin2a + sin3a

2 cosa/2 * sin3a/2 + sin3a = sina + sin2a + sin3a

sin a + sin 2a + sin 3a = sina + sin2a + sin3a