How Can You Determine the Values of ab+cd Given These Equations?

[anemone]

Here is this week's POTW:

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Suppose that 4 real numbers $a,\,b,\,c,\,d$ satisfy the conditions as shown below:

$a^2+b^2=4$
$c^2+d^2=4$
$ac+bd=2$

Evaluate all possible values for $ab+cd$.

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[anemone]

Congratulations to the following members for their correct solution::)

1. greg1313
2. kaliprasad

Solution from greg1313:

Spoiler

Let $a=2\cos(x),b=2\sin(x),c=2\cos(y),d=2\sin(y)$ then the first two conditions are satisfied.

For $ac+bd$ we have $4\cos(x)\cos(y)+4\sin(x)\sin(y)=4\cos(x-y)=2$ so we must have $x-y=\pm\dfrac{\pi}{3}+2k\pi,k\in\mathbb Z$

From all of that, $ab+cd=4\cos(x)\sin(x)+4\cos(y)\sin(y)=2(\sin(2x)+\sin(2y))=2(2\sin(x+y)\cos(x-y))=2\sin(x+y)$

Hence $-2\le ab+cd\le2$