G Vikas Vikas

As we know Complex Number are defined as
                             z = x + iy

So, Considering above, we can say the complex numbers z₁ and z₂ are
                             z₁ = x₁ + iy₁

                             z₂ = x₂ + iy₂

According to question, we need to prove,
                        Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z_2. 

Let calculate z₁z_2, we get,
                 z₁z₂ = ( x₁ + iy₁ ).( x₂ + iy₂ )

                               = x₁x₂ + ix₁y₂ + ix₂y₁ + i²y₁y₂

                               = x₁x₂ + ix₁y₂ + ix₂y₁ - y₁y₂          ( :·   i²  = -1  )

                               = x₁x₂ - y₁y₂ + ix₁y₂ + ix₂y₁                -------> (Eq 1)

Now from we can separate the real part from above Equation, we get

                   Re (z₁z₂) =  x₁x₂ - y₁y₂                  -------> (Eq 2)

Now, let take R.H.S, So we have

                  Re z₁ Re z₂ – Im z₁ Im z_2. =  x₁x₂ - y₁y₂                -------> (Eq 3)

So, Considering Eq (2) and Eq (3), we can prove that

                 Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z_2.