If `z+1//z=2costheta,` prove that `|(z^(2n)-1)//(z^(2n)+1)|=|tanntheta|`

If z=(costheta+isintheta) , show that z^(n)+(1)//(z^(n))=2cosntheta and z^(n)-(1)(z^(n))=2isinntheta

If n is a positive integer, prove that |I m(z^n)|lt=n|Im(z)||z|^(n-1.)

If |z|=1 and z'=(1+z^(2))/(z) , then

choose the correct pair of statement : (i) (costheta+isintheta)^(n)=cosntheta+isinntheta , x is an integer (ii) (sintheta+icostheta)^(n)=sinntheta+icosthetantheta , n is an integer (iii) If z=costheta+isintheta , then z^(-1)=costheta-isintheta (iv)(z+barz)/(2)=Re(z)

If |z_(1)+z_(2)|=|z_1|+|z_2| then

If z_(1)" and "z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 then

If (2z_1)/(3z_2) is purely imaginary then |(z_(1)-z_(2))/(z_(1)+z_(2))|

If z_1^2+z_2^2+2z_1.z_2.costheta= 0 prove that the points represented by z_1, z_2 , and the origin form an isosceles triangle.

Let z_(1),z_(2) and z_(3) be complex numbers such that |z_(1)|=|z_(2)|=|z_(3)|=1 then prove that |z_(1)+z_(2)+z_(3)|=|z_(1)z_(2)+z_(2)z_(3)+z_(3)z_(1)|