<p>Find the value of the following expression:<br>\[\left[\frac{1-\cos\dfrac{\pi}{10}+i\sin\dfrac{\pi}{10}}{1-\cos\dfrac{\pi}{10}-i\sin\dfrac{\pi}{10}}\right]^{10}\]</p>
Step-by-Step Solution
Key Concept: Convert the complex number to exponential form using Euler's formula and the half-angle identity: 1 - cos θ = 2sin²(θ/2). This transforms the fraction into a pure exponential form e^(iθ), which simplifies dramatically when raised to the 10th power.
<p><strong>Step 1:</strong> Apply half-angle identity: 1 - cos θ = 2sin²(θ/2)</p><p>1 - cos(π/10) = 2sin²(π/20)</p><p><strong>Step 2:</strong> Rewrite numerator and denominator using 2sin(θ/2)cos(θ/2) = sin θ</p><p>Numerator: 2sin²(π/20) + i·2sin(π/20)cos(π/20) = 2sin(π/20)[sin(π/20) + i·cos(π/20)]</p><p>= 2sin(π/20)·i[cos(π/20) - i·sin(π/20)] = 2i·sin(π/20)·e^(-iπ/20)</p><p><strong>Step 3:</strong> Denominator: 2sin(π/20)[sin(π/20) - i·cos(π/20)] = -2i·sin(π/20)·e^(iπ/20)</p><p><strong>Step 4:</strong> Form the fraction:</p><p>$$\frac{2i·sin(π/20)·e^{-iπ/20}}{-2i·sin(π/20)·e^{iπ/20}} = \frac{e^{-iπ/20}}{-e^{iπ/20}} = -e^{-iπ/10}$$</p><p><strong>Step 5:</strong> Raise to 10th power:</p><p>$$\left(-e^{-iπ/10}\right)^{10} = (-1)^{10}·e^{-iπ} = 1·(-1) = -1$$</p><p>∴ Answer: <strong>-1</strong></p>
Correct Answer: -1