Friday, 13 July 2018

trigonometry - Proving cos(w1t)+cos(w2t)=2cosleft(frac12(w1+w2)tright)cosleft(frac12(w1w2)tright)




I am trying to prove :




cos(w1t)+cos(w2t)=2cos(12(w1+w2)t)cos(12(w1w2)t)




My working so far uses the complex exponential identity:
cosθ=12(eiθ+eiθ)



12(eiw1t+eiw1t)+12(eiw2t+eiw2t)=12(eit(w1+w2)+eit(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(αβ)=cos(α)cos(β)+sin(α)sin(β) so you can add up both equalities to get cos(α+β)+cos(αβ)=2cos(α)cos(β) so now you want that α+β=ω1t and αβ=ω2t so you have to solve the system of equations α+β=ω1tαβ=ω2t
Which gives you α=12(ω1t+ω2t)β=12(ω1tω2t) which is exactly what you wanted to prove


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