<p>We have \(|z| = 1\). Then \(\dfrac{1+z}{1+\bar{z}}\) equals:</p>
Step-by-Step Solution
Key Concept: Since |z| = 1, we have z·z̄ = 1, so z̄ = 1/z. Use this reciprocal relationship to simplify the expression by multiplying numerator and denominator strategically.
<p><strong>Step 1:</strong> Given |z| = 1, this means z·z̄ = 1, so z̄ = 1/z.</p><p><strong>Step 2:</strong> Substitute z̄ = 1/z into the expression:</p><p>$$\frac{1+z}{1+\bar{z}} = \frac{1+z}{1+\frac{1}{z}}$$</p><p><strong>Step 3:</strong> Simplify the denominator:</p><p>$$\frac{1+z}{\frac{z+1}{z}} = \frac{1+z}{1} · \frac{z}{z+1} = z$$</p><p><strong>Step 4:</strong> Verification: If z = e^(iθ), then (1+e^(iθ))/(1+e^(-iθ)) = e^(iθ)·(1+e^(-iθ))/(1+e^(-iθ)) = e^(iθ) ✓</p><p>∴ Answer: <strong>z</strong></p>
Correct Answer: A