If `z_r=cos(pi/(3^r))+isin(pi/(3^r)),r=1,2,3, ,` prove that `z_1z_2z_3 .... z_oo=i`

If z=cos((pi)/(3))-isin((pi)/(3)) then z^(2)-z+1=....

If z_(r)=cos((ralpha)/(n^(2)))+isin((ralpha)/(n^(2))),"where"r=1,2,3,...,nandi=sqrt(-1),"then"lim_(n to oo) z_(1)z_(2)z_(3)...z_(n) is equal to

Let z_(1),z_(2) and z_(3) be three non-zero complex numbers and z_(1) ne z_(2) . If |{:(abs(z_(1)),abs(z_(2)),abs(z_(3))),(abs(z_(2)),abs(z_(3)),abs(z_(1))),(abs(z_(3)),abs(z_(1)),abs(z_(2))):}|=0 , prove that (i) z_(1),z_(2),z_(3) lie on a circle with the centre at origin. (ii) arg(z_(3)/z_(2))=arg((z_(3)-z_(1))/(z_(2)-z_(1)))^(2) .

If |z_(1)|=|z_(2)|=....|z_(n)|=1 , then show that, |z_(1)+z_(2)+z_(3)+....z_(n)|= |(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))+...+(1)/(z_(n))|

If z=(1)/((2+3i)^(2)) then |z|=...

If D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then prove that sum_(r=1)^n D_r=0.

For any two complex numbers z_(1) and z_(2) , prove that Re ( z_(1)z_(2)) = Re z_(1) Re z_(2)- 1mz_(1) Imz_(2)

If x=a + (a)/(r ) + (a)/(r^(2)) + …..oo, y= b - (b)/(r ) + (b)/(r^(2))-……..oo and z= c + (c )/(r^(2)) + (c )/(r^(4))+ ….oo then prove that (xy)/(z)= (ab)/(c )

If the complex numbers z_(1) and z_(2) and the origin form an isosceles triangle with vertical angle (2pi)/(3) ,then show that z_(1)^(2)+z_(2)^(2)+z_(1)z_(2)=0.