For any two complex numbers z1 and z2, prove that
Re (z1 z2) = Re z1 Re z2 – Imz1 IMz2
It is given in the question that,
z1 and z2 are two complex numbers and we have to prove that:
Re (z1z2) = Rez1 Rez2 – Imz1 Imz2
For this, firstly let z1 = x1 + iy1 and z2 = x2 + iy2
Thus, z1z2 = (x1 + iy1) (x2 + iy2)
= x1 (x2 + iy2) + iy1 (x2 + iy2)
= x1x2 + ix2y2 + iy1x2 + i2y1y2
= x1x2 + ix2y2 + iy1x2 – y1y2 (i2 = - 1)
= (x1x2 – y1y2) + i (x1y2 + y1x2)
= Re (z1z2) = x1x2 – y1y2
∴ Re (z1z2) = Rez1 Rez2 – Imz1 Imz2
Hence, proved