Answer
The equation holds. See below for proof.
Work Step by Step
We start with the left side:
$(a^{2}+b^{2})(c^{2}+d^{2})=a^{2}c^{2}+a^{2}d^{2}+b^{2}c^{2}+b^{2}d^{2}
=(a^{2}c^{2}+b^{2}d^{2}+2abcd)+(a^{2}d^{2}+b^{2}c^{2}-2abcd)=(ac+bd)^{2}+(ad-bc)^{2}$
Which equals the right side.
