Step-by-Step Solution
Key Concept: Express e^(iθ) in Euler form as cosθ + isinθ, then e^(e^(iθ)) = e^(cosθ + isinθ) = e^(cosθ) · e^(isinθ), and extract the real part using Euler's formula for e^(isinθ).
<p><strong>Step 1:</strong> Apply Euler's formula to the exponent: e^(iθ) = cosθ + isinθ</p><p><strong>Step 2:</strong> Substitute into e^(e^(iθ)): e^(e^(iθ)) = e^(cosθ + isinθ)</p><p><strong>Step 3:</strong> Separate the exponents: e^(cosθ + isinθ) = e^(cosθ) · e^(isinθ)</p><p><strong>Step 4:</strong> Apply Euler's formula to e^(isinθ): e^(isinθ) = cos(sinθ) + i·sin(sinθ)</p><p><strong>Step 5:</strong> Multiply: e^(cosθ)[cos(sinθ) + i·sin(sinθ)]</p><p><strong>Step 6:</strong> The real part is the coefficient of the non-imaginary term.</p><p>∴ Real part of e^(e^(iθ)) = <strong>e^(cosθ)cos(sinθ)</strong></p>
Correct Answer: e^(cosθ)[cos(sinθ)]