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Given that z + 1/z = 2*cos(theta), find the value of the positive integer k such that |(z^(2n) - 1) / (z^(2n) + 1)| = k * |tan(n*theta)| holds for all valid z and integer n.

1. 1
2. 2
3. 3
4. 4

Correct answer: 1

Solution

With z = e^(i*theta), z^(2n) = e^(2ni*theta). Multiplying numerator and denominator by e^(-ni*theta) gives (e^(ni*theta) - e^(-ni*theta))/(e^(ni*theta) + e^(-ni*theta)) = i*tan(n*theta), whose modulus is |tan(n*theta)|. Hence k = 1.

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