Find two solutions to solve sin15 + sin75 .

One method would be to consider the fact that being an addition of two alike functions, we'll transform the addition into a product, in this way:

sin a + sin b = 2sin [(a+b)/2]cos [(a-b)/2]

sin 15 + sin 75 = 2sin[(15+75)/2] cos [(15-75)/2]

sin 15 + sin 75 = 2sin45cos30

sin 15 + sin 75 = 2*[(sqrt2)/2]*[(sqrt3)/2]

sin 15 + sin 75 = sqrt(2*3)/2=sqrt(6)/2

Another manner of solving would be to write the angles:

15 = 45 - 30

75 = 45 + 30

sin (45 - 30) = sin45*cos30 - sin30*cos45

                  =(sqrt2/2)(sqrt3/2) - sqrt2/4

                  = (sqrt6 - sqrt2)/4

sin (45 + 30) = sin45*cos30 + sin30*cos45

                  = (sqrt6 + sqrt2)/4

So,

sin 15 + sin 75 = (sqrt6 - sqrt2+sqrt6 + sqrt2)/4

sin 15 + sin 75 = 2sqrt6/4

sin 15 + sin 75 = sqrt6/2