# For any two complex numbers z1 and z2, prove thatRe (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

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