Trigomplex

consider \\ S=e^{i2x}+e^{i4x}+e^{i6x}+.........+e^{i180x} which \ is \ a \ g.p \\ \frac{dS}{dx}=S' \\ answer = \boxed {Im(S')/(2+4+6+...180)}

2 sin(2) + 178 sin(178) = 180 sin 2
4 sin (4) + 176 sin(176) = 180 sin 4
6 sin (6) + 174 sin (176) = 180 sin 6
..............
..............
..............
90 sin (90) + 90 sin (180 - 90 ) = 180 sin 90

so avg. = [ sin 2 + sin 4 + sin 6 + sin 8 + .......+ sin 90180 ] x 180

Now . sin a + sin(a+d) + sin (a+2d ) + ...... + sin (a+nd ) = sin (na /2 ) sin ( a + {n-1} d/2 )sin (d/2)

Compaaring with the given series , a = 2 , d =2 , n = 2

so avg. = sin 90 sin ( 2 + 89 )sin 1 = cot 1

hmm....u made a mistake i guess in taking average

i was looking for something through complex