# Prove this identity:(a^2 + b^2) (c^2 + d^2) = (ac +\- bd)^2 + (ad -\+ bc)^2

## Expert Answers

(a^2 + b^2) (c^2 + d^2) = (ac +- bd)^2 + (ad -+ bc)^2

Let us start from the right side:

(ac + bd)^2 + (ad - bc)^2 = (ac)^2 + 2acbd + (bd)^2 + (ad)^2 - 2abcd + (bc)^2

Let us simplify:

(ax+bd)^2 + (ad-bc)^2 = (ac)^2 + (bd)^2...

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(a^2 + b^2) (c^2 + d^2) = (ac +- bd)^2 + (ad -+ bc)^2

Let us start from the right side:

(ac + bd)^2 + (ad - bc)^2 = (ac)^2 + 2acbd + (bd)^2 + (ad)^2 - 2abcd + (bc)^2

Let us simplify:

(ax+bd)^2 + (ad-bc)^2 = (ac)^2 + (bd)^2 + (ad)^2 + (bc)^2

Let us rearrange terms:

==> (ac+ bd)^2 + (ad-bc)^2= (ac)^2 (bc)^2 + (bd)^2 + (ad)^2

Now we will factor:

= c^2 (a^2+b^2) + d^2(a^2+b^2)

= (a^2+ b^2)(c^2 + d^2)...

==> (ac+bd)^2 + (ad-ac)^2 = (a^2 + b^2 )(c^2 + d^2)

Similarly:

(ac-bd)^2 + (ad+ac)^2 = (a^2 + b^2)((c^2 + d^2)

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