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intresting question

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It’s rather elementary to prove this from the definitions.

“Can I express the sine of a sum in terms of the trig functions of the original addends?” is something very natural to wonder.

Then you can just draw a right triangle with angle measure x+y and a line that splits it into angles of measure x and y then roll with some basic geometry until you get an expression for sin(x+y).
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My philosophy on "how did people figure this out back in the day?" is that folks used to be *bored* for most of their downtime. Less immediate, novel stimulation = less dopamine = more experimentation. If Necessity is the mother of Invention, then Boredom is the father.
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You have x + y inside the sin, you want to separate them so they're two separate additive terms. It's natural then that you would discover this identity.
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Try Euler's identity e^iz =isinz+cosz and the working will flow naturally just substitute X+y into z and use the indices rules to separate them and both the sin and cosine will be solved .
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Almost all trig identities are easier to prove if you use Euler’s identity.

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