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For any two complex numbers z 1 and z 2 , prove that Re (z1z2) = Re z1 Re z2 - Im z1 Im z2

Q : 2       For any two complex numbers  \small z_1  and  \small z_2,  prove that 

                \small Re (z_1z_2)=Re\hspace {1mm}z_1\hspace {1mm}Re\hspace {1mm}z_2-Imz_1\hspace {1mm}Imz_2

Answers (1)
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Let two complex numbers are
z_1=x_1+iy_1
z_2=x_2+iy_2
Now,
z_1.z_2=(x_1+iy_1).(x_2+iy_2)
            =x_1x_2+ix_1y_2+iy_1x_2+i^2y_1y_2                               
            =x_1x_2+ix_1y_2+iy_1x_2-y_1y_2                                        (\because i^2 = -1)
            =x_1x_2-y_1y_2+i(x_1y_2+y_1x_2)
Re(z_1z_2)= x_1x_2-y_1y_2
                    =Re(z_1z_2)-Im(z_1z_2)

Hence proved

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