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Question: Real part of \(e^{e^{i\theta}}\) is...

Real part of eeiθe^{e^{i\theta}} is

A

ecos⁡θ[cos⁡(sin⁡θ)]e^{\cos\theta}\lbrack\cos(\sin\theta)\rbrack

B

ecos⁡θ[cos⁡(cos⁡θ)]e^{\cos\theta}\lbrack\cos(\cos\theta)\rbrack

C

esin⁡θ[sin⁡(cos⁡θ)]e^{\sin\theta}\lbrack\sin(\cos\theta)\rbrack

D

esin⁡θ[sin⁡(sin⁡θ)]e^{\sin\theta}\lbrack\sin(\sin\theta)\rbrack

Answer

ecos⁡θ[cos⁡(sin⁡θ)]e^{\cos\theta}\lbrack\cos(\sin\theta)\rbrack

Explanation

Solution

Sol.eeiθ=e(cos⁡θ+isin⁡θ)=ecos⁡θ.eisin⁡θ=ecos⁡θ[cos⁡(isin⁡θ)+isin⁡(sin⁡θ)]e^{e^{i\theta}} = e^{(\cos\theta + i\sin\theta)} = e^{\cos\theta}.e^{i\sin\theta} = e^{\cos\theta}\lbrack\cos(i\sin\theta) + i\sin(\sin\theta)\rbrack

∴ Real part of eeiθe^{e^{i\theta}}is ecos⁡θ[cos⁡(sin⁡θ)]e^{\cos\theta}\lbrack\cos(\sin\theta)\rbrack.