if `omega` is the nth root of unity and `Z_1 , Z_2 ` are any two complex numbers , then prove that .
`Sigma_(k=0)^(n-1)| z_1+ omega^k z_2|^2=n{|z_1|^2+|z_2|^2}` where `n in N `

If 1,alpha,alpha^(2),………..,alpha^(n-1) are the n, n^(th) roots of unity and z_(1) and z_(2) are any two complex numbers such that sum_(r=0)^(n-1)|z_(1)+alpha^(R ) z_(2)|^(2)=lambda(|z_(1)|^(2)+|z_(2)|^(2)) , then lambda=

For any two 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]

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

If z_1a n dz_2 are two complex numbers and c >0 , then prove that |z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot

For any two complex number z_1a n d\ z_2 prove that: |z_1+z_2|geq|z_1|-|z_2|

For any two complex number z_1a n d\ z_2 prove that: |z_1-z_2|geq|z_1|-|z_2|

For any two complex number z_1a n d\ z_2 prove that: |z_1-z_2|lt=|z_1|+|z_2|

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2=|\z_1|^2+|\z_2|^2+2Re bar z_1 z_2

For any two complex numbers z_1 and z_2 , , prove that Re (z_1 z_2) = Re z_1 Re z_2 – Imz_1 Imz_2

For any two complex numbers z_1 and z_2 , prove that Re(z_1z_2)=Re(z_1) Re(z_2)-Im(z_1) Im(z_2) .