Complex Numbers

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If z = cos θ + i sin θ, then z^n + 1/z^n equals

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2 cos(nθ)

2 sin(nθ)

2i cos(nθ)

2i sin(nθ)

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About This Question

Subject: mathematics
Chapter: complex numbers and quadratic equations
Topic: complex numbers
Difficulty: Medium
Year: 2025

Solution

Correct Answer:

2 cos(nθ)

De Moivre's Theorem: $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$ .
Express Terms:
- $z^{n} = cos (n θ) + i sin (n θ)$
- $\frac{1}{z ^{n}} = z^{- n} = cos (- n θ) + i sin (- n θ) = cos (n θ) - i sin (n θ)$
Addition:
- $z^{n} + z^{- n} = (cos n θ + i sin n θ) + (cos n θ - i sin n θ)$
- $= 2 cos (n θ)$ .

This medium difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of complex numbers. It appeared in the 2025 exam.

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