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Here is a geometric derivation of the addition laws for sines (and cosines)
First note that, by symmetry, it is clear that
and so we can reduce proving the addition law to proving it in the case where , , and
are all in the first quadrant.
We then have the situation pictured below:
Now, from the definitions of and , and assuming that , we see that
Now,
so we have that
But also, it is clear that
, and therefore we have similarly
so that
Thus
is the -coordinate of , which is the -coordinate of , so
and
is the -coordinate of , which is the -coordinate of less the difference in -coordinates between and , so
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