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

If z + (1)/(z) = 2 cos theta, z in "C then z"^(2n) - 2z^(n) cos (n theta) is equal to

If z_(1) and z_(2) two different complex numbers and |z_(2)|=1 then prove that |(z_(2)-z_(1))/(1-bar(z)_(1)z_(2))|=1

If |z_1|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.

If z_(1) and z_(2) are two complex numbers such that |z_(1)| lt 1 lt |z_(2)| , then prove that |(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1

For any two complex numbers z_(1) and z_(2), prove that |z_(1)+z_(2)| =|z_(1)|-|z_(2)| and |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

If z_(1) and z_(2) are two complex number such that |z_(1)|<1<|z_(2)| then prove that |(1-z_(1)bar(z)_(2))/(z_(1)-z_(2))|<1

If z_(1) and z_(2) are two complex numbers then prove that |z_(1)-z_(2)|^(2)<=(1+k)z_(1)^(2)+(1+(1)/(k))z_(2)^(2)

if |z_(1)+z_(2)|=|z_(1)|+|z_(2)|, then prove that arg(z_(1))=arg(z_(2)) if |z_(1)-z_(2)|=|z_(1)|+|z_(2)| then prove that arg (z_(1))=arg(z_(2))=pi

For all complex numbers z_(1) and z_(2) , prove that |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2)=2(|z_(1)|^(2)+|z_(2)|^(2)) .