SOLUTION: 1) use Euler's formula to determine the standard trigonometric identity for cos(alpha+beta).

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Question 114238: 1) use Euler's formula to determine the standard trigonometric identity for cos(alpha+beta).
Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
Euler's Formula is:

e%5E%28ix%29=cos%28x%29%2Bi%2Asin%28x%29

Apply this formula to x=alpha%2Bbeta

e%5E%28i%28alpha%2Bbeta%29%29=cos%28alpha%2Bbeta%29%2Bi%2Asin%28alpha%2Bbeta%29

But e%5E%28i%28alpha%2Bbeta%29%29=e%5E%28i%2Aalpha%29e%5E%28i%2Abeta%29

So

Which is to say,

Expanding the right side using FOIL yields:


And gathering like terms:


Since we know that a%2Bbi=c%2Bdi if and only if a=c and b=d,
we can see that:

cos%28alpha%2Bbeta%29=cos%28alpha%29cos%28beta%29-sin%28alpha%29sin%28beta%29, which is the desired identity.

By the way, we have also proven the companion identity:

sin%28alpha%2Bbeta%29=cos%28alpha%29sin%28beta%29%2Bsin%28alpha%29cos%28beta%29

Hope this helps,
John