For any two complex numbers $z_{1}$ and $z_{2}$ , prove that
$R e (z_{1} z_{2}) = R e z_{1} R e z_{2} - I m z_{1} I m z_{2}$ .

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Solution

Let $z_{1} = a + i b z_{2} = x + i y$
$z_{1} z_{2} = (a + i b) (x + i y) a x - b y + i (b x + a y) R e (z_{1} z_{2}) = a x - b y R e (z_{1} z_{2}) = R e z_{1} R e z_{2} - I m z_{1} I m z_{2}$

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For any two complex numbers z1 and z2, prove that Re(z1z2)=Rez1Rez2−Imz1Imz2.

Let z1=a+ibz2=x+iyz1z2=(a+ib)(x+iy)ax−by+i(bx+ay)Re(z1z2)=ax−byRe(z1z2)=Rez1Rez2−Imz1Imz2

For any two complex numbers $z_{1}$ and $z_{2}$ , prove that
$R e (z_{1} z_{2}) = R e z_{1} R e z_{2} - I m z_{1} I m z_{2}$ .

Let $z_{1} = a + i b z_{2} = x + i y$
$z_{1} z_{2} = (a + i b) (x + i y) a x - b y + i (b x + a y) R e (z_{1} z_{2}) = a x - b y R e (z_{1} z_{2}) = R e z_{1} R e z_{2} - I m z_{1} I m z_{2}$