The realization that there is a problem with a correlation dependent on -cos(θ) may not be intuitively obvious. Which is probably why it took 10+ years after Bohm's work on spin entanglement in the early 1950's for someone like Bell to figure it out.

Theta is a difference between 2 measurement angles. Because it is not straight line linear for the difference, at some point you realize that the dependence is actually contextual - based only on a future pair of measurement angles. In other words, you would expect a theta of 45 degrees to yield exactly a value of twice a theta of 22.5 degrees (for example). But that doesn't happen. Instead, the relationship changes much faster. Matches at 22.5 degrees are about 3.8%, but at 45 degrees are about 14.6%. The Matches from 0 to 22.5 degrees plus the Matches from 22.5 to 45 degrees should be the same (or less) as from 0 to 45 degrees.

It may take a while to see why this is so. But usually, references to the cosine or sine relationship for theta is based on this general argument. Of course, it is even better to follow the Bell argument first so that you can see specific relationships that cannot be possible under standard QM (but would be possible under local realism).

Or, to answer your question a different way: nothing looks amiss which you only look at 2 angles and use -cos(theta) for Correlation. It is when you start comparing hypothetical values for 3 or more values that you see the relationships won't ever work out.

And if you read Bell, he essentially uses your example as his starting point. And then develops his contradictions from there. See Bell 1964: https://www.informationphilosopher.com/solutions/scientists/bell/Bell_On_EPR.pdf formulae (6) and (7).

The realization that there is a problem with a correlation dependent on -cos(θ) may not be intuitively obvious. Which is probably why it took 10+ years after Bohm's work on spin entanglement in the early 1950's for someone like Bell to figure it out.

Theta is a difference between 2 measurement angles. Because it is not straight line linear for the difference, at some point you realize that the dependence is actually contextual - based only on a future pair of measurement angles. In other words, you would expect a theta of 45 degrees to yield exactly a value of twice a theta of 22.5 degrees (for example). But that doesn't happen. Instead, the relationship changes much faster. Matches at 22.5 degrees are about 3.8%, but at 45 degrees are about 14.6%. The Matches from 0 to 22.5 degrees plus the Matches from 22.5 to 45 degrees should be the same (or less) as from 0 to 45 degrees.

It may take a while to see why this is so. But usually, references to the cosine or sine relationship for theta is based on this general argument. Of course, it is even better to follow the Bell argument first so that you can see specific relationships that cannot be possible under standard QM (but would be possible under local realism).

Or, to answer your question a different way: nothing looks amiss which you only look at 2 angles and use -cos(theta) for Correlation. It is when you start comparing hypothetical values for 3 or more values that you see the relationships won't ever work out.

If there are hidden variables, then there should be relationships that work out for all possible values - simultaneously. The "simultaneous" condition was assumed in EPR (1935). But if you read Bell, he essentially uses your example as his starting point. See Bell 1964: https://www.informationphilosopher.com/solutions/scientists/bell/Bell_On_EPR.pdf formulae (6) and (7). And then develops his contradictions from there, explicitly modeling simultaneous relationships.