`z^n-1=0 =>\ \ \ \ \ \ z=(1)^(1/n)=(cos0+isin0)^(1/n)=(cos2mpi+i sin 2mpi)^(1/n)`

`=>cos((2mpi)/n)+i sin ((2mpi)/n) ,\ \ \ \ m=0, 1, 2 , 3 ,............,(n-1)`

`m = 0, z_1 = 1`

`m=1 , z_2= cos((2pi)/n)+i sin ((2pi)/n)= e^(i(2pi)/n)=alpha(Let)`

`m=2 , z_3= cos((4pi)/n)+i sin ((4pi)/n)= e^(i(4pi)/n)=alpha^2(Let)`
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`m= (n - 1), z_n = cos((2(n-1)pi)/n)+i sin ((2(n-1)pi)/n)= e^(i((2(n-1)pi)/n))=alpha^(n-1)(Let)`

`text(Important Results : )`

`1.` If `2, alpha_1, alpha_2,alpha_3,..........,alpha_(n-1)` nth roots of unity, then

`(1)^p + (alpha_1)^p + (alpha_2)^p + ......+ (alpha_(n-1))^p = { tt[(0, text{if pis not an integral multiple of n}),(n,text{1f p is an mtegral multiple of n})]`

`2. (1 + alpha_1)(1-alpha_2).............(1- alpha_(n-1)) = { tt[(0, text{if n is even}),(1,text{if n is odd})]`

`3.(1-alpha_1)(1-alpha_2)..........(1-alpha_(n-1)) = n`

`4. z^n -1 = (z-1)(z+1) Pi_(r=1)^(n-2)//2(z^2 - 2z cos (2rpi)/n + 1)` if 'n' is even

`5.z^n +1 = Pi_(r=1)^((n-2)//2)(z^2-2z cos(((2r+1)pi)/pi)+1),` if n even

`6.z^n +1 = Pi_(r=0)^((n-3)//2)(z^2-2z cos(((2r+1)pi)/pi)+1),` if 'n' odd


`text(The Sum of the Following Series Should be Remembered : )`

`(i) cos theta + cos 2 theta + cos 3 theta + .........+ cos n theta = (sin((n theta)/2))/(sin(( theta)/2)) . cos[((n+1)/2)theta]`

`(ii) sin theta + sin 2 theta + sin 3 theta + .........+ cos n theta = (sin((n theta)/2))/(sin(( theta)/2)) . sin[((n+1)/2)theta]`