TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Finite binomial formulas for $\cos (n θ)$ and $\sin (n θ)$

Statement

For every $n \in N$ and real $θ$ , $\cos (n θ) = \sum_{2 j \leq n} (- 1)^{j} (\binom{n}{2 j}) \cos^{n - 2 j} θ \sin^{2 j} θ,$ $\sin (n θ) = \sum_{2 j + 1 \leq n} (- 1)^{j} (\binom{n}{2 j + 1}) \cos^{n - 2 j - 1} θ \sin^{2 j + 1} θ .$ The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Pascal's rule $(\binom{n + 1}{k + 1}) = (\binom{n}{k}) + (\binom{n}{k + 1})$ , and the hockey-stick identity $\sum_{i \leq n} (\binom{i}{k}) = (\binom{n + 1}{k + 1})$ , The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , Finite sums and finite products, by recursion, Integer powers $a^{m}$ , The principle of mathematical induction.

Facts & Assumptions

Given: A natural $n$ and real $θ$ .

[L1]

The addition formulas for sine and cosine gives the formulas for $\sin (a + b)$ and $\cos (a + b)$ for all real $a, b$ .

[L2]

Pascal's rule $(\binom{n + 1}{k + 1}) = (\binom{n}{k}) + (\binom{n}{k + 1})$ , and the hockey-stick identity $\sum_{i \leq n} (\binom{i}{k}) = (\binom{n + 1}{k + 1})$ gives $(\binom{n + 1}{k + 1}) = (\binom{n}{k}) + (\binom{n}{k + 1})$ for all natural $n, k$ .

Proof

technique · induction

1.1

At $n = 0$ the two displayed finite sums give $1$ and $0$ .

base

1.2

Assume the two formulas at $n$ .

2.1

Apply [L1] to $n θ + θ$ and insert the two induction sums. Collecting the coefficient of each monomial $\cos^{n + 1 - r} θ \sin^{r} θ$ leaves the sum of the two adjacent binomial coefficients.

L1step 1.2algebra

3.1

By [L2], those adjacent sums are exactly $(\binom{n + 1}{r})$ ; even $r$ contribute to cosine with sign $(- 1)^{r / 2}$ and odd $r$ contribute to sine with sign $(- 1)^{(r - 1) / 2}$ . This proves both formulas at $n + 1$ without using complex numbers.

L2step 2.1discharge-induction∎

Depends on

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Sources

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)

Finite binomial formulas for cos⁡(nθ)cos(nθ) and sin⁡(nθ)sin(nθ)