1U can use complex nos. as well. that is much faster.
The common diff. shud give the hint.
sin (a ) + sin (a + 2pi/3 ) + sin ( 4pi/3) is nothing but the imaginary part of cube roots of unity which have rotated by an angle 'a'. Hence the sum of the img ( and the real part shud be zero.
Using vectors too, these are the y - component of three unit modulus vectors making an angle 2pi/3 with each other. Hence the net vector is zero and hence the y and x components are independently adding upto zero.
Another good question involving same concept is
If p, g , r are the sine(s) of angle in AP with cd = 2pi/3, then find
p2 + q2 + r2 =
Keep in mind that here u are considerin squares and hence the components won't cancel each other to give zero.
trick here is to use the double - angle formula to convert it into
1/2{cos (2a ) + cos (2a + 4 pi/3 ) + cos ( 2a + 8pi/3)} + 3/2 = 3/2
as the cos terms again add to zero by the same method used above.