I am trying to prove :
cos(w1t)+cos(w2t)=2cos(12(w1+w2)t)cos(12(w1−w2)t)
My working so far uses the complex exponential identity:
cosθ=12(eiθ+e−iθ)
12(eiw1t+e−iw1t)+12(eiw2t+e−iw2t)=12(eit(w1+w2)+e−it(w1+w2))=cos((w1+w2)t)=cos(w1t+w2t)
I cannot understand how to proceed further. Please help.
Answer
Remember that cos(α+β)=cos(α)cos(β)−sin(α)sin(β) and cos(α−β)=cos(α)cos(β)+sin(α)sin(β) so you can add up both equalities to get cos(α+β)+cos(α−β)=2cos(α)cos(β) so now you want that α+β=ω1t and α−β=ω2t so you have to solve the system of equations α+β=ω1tα−β=ω2t
Which gives you α=12(ω1t+ω2t)β=12(ω1t−ω2t) which is exactly what you wanted to prove
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