for any two complex no. z1 and z2 prove that re(z1 z2) = re(z1) * re(z2) - im(z1) * im(z2)
my incomplete soln-
don't have any idea about it.
$\displaystyle \displaystyle z_1 = x_1 + iy_1, z_2 = x_2 + iy_2$
$\displaystyle \displaystyle \begin{align*}z_1z_2 &= (x_1 + iy_1)(x_2 + iy_2) \\ &= x_1x_2 + ix_1y_2 + ix_2y_1 + i^2y_1y_2 \\ &= x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)\end{align*}$
So what is the real part of $\displaystyle \displaystyle z_1z_2$?
You can write any complex number as $\displaystyle \displaystyle Z = X + iY$, where $\displaystyle \displaystyle X$ is the real part of $\displaystyle \displaystyle Z$ and $\displaystyle \displaystyle Y$ is the imaginary part of $\displaystyle \displaystyle Z$.
You have $\displaystyle \displaystyle z_1z_2 = x_1x_2 - y_1y_2 + i(x_1y_2 + x_2y_1)$. If you're going to rewrite this as $\displaystyle \displaystyle Z= X + iY$, what is $\displaystyle \displaystyle X$ (the real part of $\displaystyle \displaystyle z_1z_2$)?
Because $\displaystyle \displaystyle x_1$ is the real part of $\displaystyle \displaystyle z_1$, $\displaystyle \displaystyle x_2$ is the real part of $\displaystyle \displaystyle z_2$, $\displaystyle \displaystyle y_1$ is the imaginary part of $\displaystyle \displaystyle z_1$ and $\displaystyle \displaystyle y_2$ is the imaginary part of $\displaystyle \displaystyle z_2$.