I am stuck on trying to prove a trig identity using De Moivre's theorem.
I have to prove, cos(3θ)=4cos3(θ)−3cos(θ)
I am not sure where to even start, I broke the LHS down to cos(3θ)+isin(3θ)
but I have no idea where to go from here, or if this is fully correct.
If I could get some pointers or a simple worked example that I could follow it would be great.
Thanks
Answer
De Moivre's formula reads (cosθ+isinθ)n=cos(nθ)+isin(nθ) Of course this identity implies the real part should be also equality. That is cos(nθ)=ℜ{(cosθ+isinθ)n} Hence we have cos(3θ)=ℜ{cos3θ+3icos2θsinθ−3cosθsin2θ−isin3θ}=cos3θ−3cosθsin2θ
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