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If alpha = pi/12, then the value of ((cos(alpha) + i*sin(alpha)) * (cos(2*alpha) + i*sin(2*alpha))) / (cos(3*alpha) - i*sin(3*alpha)) is

1. 0
2. -i
3. -1
4. i

Correct answer: i

Solution

The numerator equals e^(i*alpha) * e^(i*2*alpha) = e^(i*3*alpha). The denominator equals e^(-i*3*alpha). So the full expression equals e^(i*3*alpha) / e^(-i*3*alpha) = e^(i*6*alpha). With alpha = pi/12, we get 6*alpha = pi/2. Therefore e^(i*pi/2) = cos(pi/2) + i*sin(pi/2) = 0 + i = i.

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