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

If Z_r=cos((2rpi)/5)+isin((2rpi)/5),r=0,1,2,3,4,... then z_1z_2z_3z_4z_5 is equal to

If z_n=[cos (pi/2)^(n/2)+i sin (pi/2)^(n/2)] then z_1.z_2.z_3…….oo

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

If xcostheta=ycos(theta+(2pi)/3)=z cos(theta+(4pi)/3) , prove that x y+y z+z x=0.

If xcostheta=ycos(theta+(2pi)/3)=z cos(theta+(4pi)/3) , prove that x y+y z+z x=0.

If z=cos(pi/4)+isin(pi/6) , then a. |z|=1, arg(z)=\ pi/4 b. |z|=1, arg(z)=\ pi/6 c. |z|=(sqrt(3))/2, arg(z)=(5pi)/(24) d. |z|=(sqrt(3))/2, arg(z)=tan^(-1)(1/sqrt(2))

If |z|=2a n d(z_1-z_3)/(z_2-z_3)=(z-2)/(z+2) , then prove that z_1, z_2, z_3 are vertices of a right angled triangle.

If z_1=i z_2 and z_1-z_3=i(z_3-z_2) ,then prove that |z_3|=sqrt(2)|z_1|.

Let z_(1),z_(2) and z_(3) be three complex number such that |z_(1)-1|= |z_(2) - 1| = |z_(3) -1| and arg ((z_(3) - z_(1))/(z_(2) -z_(1))) = (pi)/(6) then prove that z_(2)^(3) + z_(3)^(3) + 1 = z_(2) + z_(3) + z_(2)z_(3) .

If |z_1|=|z_2|=.......=|z_n|=1, prove that |z_1+z_2+z_3++z_n|=1/(z_1)+1/(z_2)+1/(z_3)++1/(z_n)