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If a complex number z satisfies z + 1/z = 2cosθ, then the magnitude of (z^(2n) - 1)/(z^(2n+1) + 1) is

1. |tan nθ|
2. tan nθ
3. |cot nθ|
4. cot nθ

Correct answer: |tan nθ|

Solution

With z + 1/z = 2cosθ we have z = e^(iθ). Then (z^(2n) - 1)/(z^(2n) + 1) = (e^(inθ)·2i·sin nθ)/(e^(inθ)·2·cos nθ) = i·tan nθ, whose magnitude is |tan nθ|. Since a modulus cannot be negative, the answer is |tan nθ|.

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