Question 2:

For any two complex numbers z₁ and z₂, prove that

Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Answer

Let z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂

∴ z₁z₂ = (x₁ + iy₁)(x₂ + iy₂)              

             = x₁(x₂ + iy₂) + iy₁­(x₂ + iy₂)              

             = x₁x₂ +ix₁y₂ + iy₁x₂ + i²y₁y­₂              

             = x₁x₂ + ix₁y₂ + iy₁x₂ + i²y₁y₂              

             = x₁x₂ + ix₁y₂ + 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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