<s>[INST] How do you show $(ac)^2+(bd)^2+(bc)^2+(ad)^2=1$ Show $(ac)^2+(bd)^2+(bc)^2+(ad)^2=1$ if $a^2+b^2=1$ and $c^2+d^2=1$.
I was thinking about solving for variables and plugging in, but that seems like too much work. Is there a simple trick or realization that will help me prove this more quickly?
 [/INST] \begin{align}
& (a^2×c^2)+(b^2×d^2)+(b^2×c^2)+(a^2×d^2) \\
= {} & (a^2×c^2)+(b^2×c^2)+(b^2×d^2)+(a^2×d^2) \\
= {} & c^2(a^2+b^2) + d^2(a^2+b^2) \\
= {} & (a^2+b^2) (c^2+d^2) \\
= {} & 1\times1 \\
= {} & 1
\end{align}
Hope that makes more sense!
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