inspiration:- Mistakes give rise to problems!

Geometry Level 4

\[\large\sin(x+y)=\sin(x)+\sin(y)\] We know that the above splitting is not allowed. This is a false property, it will be a mistake if you do it! But for how many ordered pairs \((x,y)\), where \(x\) and \(y\) are integral multiples of \(\pi\) and \(-\pi\leq x,y\leq 3\pi\), does the above equality seem to hold true?

Inspired by Aditya Raut's set Mistakes Give Rise To Problems!.
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