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De Moivre's formula using prove that,

If a=cosθ+isinθ,b=cosϕ+isinϕ prove that

(i) cos(θ+ϕ)=12[ab+1ab]

(ii) sin(θϕ)=12i[abba]

1 Answer
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Solution:

Given, a=cosθ+isinθ

&  b=cosϕ+isinϕ

(i) cos(θ+ϕ)=12[ab+1ab] prove that,

Now, ab=(cosθ+isinθ)(cosϕ+isinϕ)

ab=cos(θ+ϕ)+isin(θ+ϕ)

also,1ab=(ab)1=[cos(θ+ϕ)+isin(θ+ϕ)] $ \Rightarrow \frac{1}{a b}=\cos (\theta+\phi)-i \sin …

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