How do I transform the left side into the right side of this equation? How does one transform the left side into the right side?
$$
(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2
$$

$(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2$
If $(a^2+b^2)\ne 0$ and $(c^2+d^2)\ne 0$ (If either of them is $0$ then the statement is vacuously true).
Let $\sin \alpha =\displaystyle \frac{a}{\sqrt{a^2+b^2}}\Rightarrow \cos \alpha\displaystyle \frac{b}{\sqrt{a^2+b^2}}$ and $\sin \beta =\displaystyle \frac{c}{\sqrt{c^2+d^2}}\Rightarrow \cos \beta\displaystyle \frac{d}{\sqrt{c^2+d^2}}$
So we have ,
$\displaystyle \frac{(ac-bd)^2 + (ad+bc)^2}{(a^2+b^2)(c^2+d^2)}=(-\cos (\alpha+\beta))^2+(\sin(\alpha+\beta))^2 =1$ 
We are done.
