Exercise 1538Combinatorics › Binomial coefficientsOlympiadAdded 3 Nov 2026

A complex binomial identity

#complex numbers#binomial#identity

Statement

Compute $\sum_{k=0}^{n}\binom{n}{k}\cos(k\theta)$ using complex numbers.

Hint

Take the real part of $(1+e^{i\theta})^n$ .

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Complex. $(1+e^{i\theta})^n=\sum_k\binom{n}{k}e^{ik\theta}$ .

Real part. $\sum_k\binom{n}{k}\cos(k\theta)=\operatorname{Re}((1+e^{i\theta})^n)$ .

Compute. $1+e^{i\theta}=2\cos(\theta/2)e^{i\theta/2}$ , so $(1+e^{i\theta})^n=2^n\cos^n(\theta/2)e^{in\theta/2}$ .

Conclusion. $\sum_k\binom{n}{k}\cos(k\theta)=2^n\cos^n(\theta/2)\cos(n\theta/2)$ .