For any two complex numbers z1&z2 prove that Re(z1z2) =Rez1Rez2-Imz1Imz2

Question

this question is from chapter 5 ,complex numbers, Book NCERT, micellaneous exercises, page no.112.

Answer 1

Let z1=x+iy and z2=a+ib. Therefore Re(z1)=x and Re(z2)=a -------------(i) and Im(z1)=y and Im(z2)=b -----------(ii) then we get z1z2=(x+iy)*(a+ib) =x*(a+ib)+iy*(a+ib) =ax+ibx+iay+(i*i)by =ax+i(bx+ay)-by, since i^2=-1 =(ax-by)+i(bx+ay) Therefore we get Re(z1z2)=ax-by Re(z1z2)=Re(z1)Re(z2)-Im(z1)Im(z2), using (i) and (ii). (Proved)