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Proof of Product-to-Sum Formulas for Sine and Cosine

lemmalem:trigonometric-product-formulas-2026a
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· 2,548 chars · 4 deps · depth 15 Reason: First publication. The addition formulas evaluated at $-v$, together with the parity of sine and cosine, give the formulas for a difference; adding and subtracting these against the formulas for a sum yields the three identities.

The addition formulas applied at −v-v, together with the parity of sine and cosine, give the formulas for a difference; adding and subtracting them against the formulas for a sum produces the three identities.

Proof

Each result cited below is universally quantified over the data in its own statement; it is applied to the data named at the point of use. Throughout, R\mathbb{R} is a field, so its addition and multiplication are commutative and associative, multiplication distributes over addition, and s+(−s)=0s+(-s)=0; these are used below in rearranging the finite sums that occur. Let u,v∈Ru,v\in\mathbb{R}.

Claim 1 (the formulas for a difference).

cos⁡(u−v)=cos⁡ucos⁡v+sin⁡usin⁡v,sin⁡(u−v)=sin⁡ucos⁡v−cos⁡usin⁡v.\cos(u-v)=\cos u\cos v+\sin u\sin v, \qquad \sin(u-v)=\sin u\cos v-\cos u\sin v .

Since u−v=u+(−v)u-v=u+(-v), clause Addition Formulas for Sine and Cosine §cosine, applied with a=ua=u and x=−vx=-v, gives

cos⁡(u−v)=cos⁡ucos⁡(−v)−sin⁡usin⁡(−v),\cos(u-v)=\cos u\cos(-v)-\sin u\sin(-v),

and clause Addition Formulas for Sine and Cosine §sine gives

sin⁡(u−v)=sin⁡ucos⁡(−v)+cos⁡usin⁡(−v).\sin(u-v)=\sin u\cos(-v)+\cos u\sin(-v).

By Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine §parity, cos⁡(−v)=cos⁡v\cos(-v)=\cos v and sin⁡(−v)=−sin⁡v\sin(-v)=-\sin v. Claim 2 of Zero Products and Elementary Identities in a Field gives s(−r)=−(sr)s(-r)=-(sr) for all s,r∈Rs,r\in\mathbb{R}, so −sin⁡usin⁡(−v)=sin⁡usin⁡v-\sin u\sin(-v)=\sin u\sin v and cos⁡usin⁡(−v)=−(cos⁡usin⁡v)\cos u\sin(-v)=-(\cos u\sin v), which yields the two displayed identities.

Claim 2 (doubling). For every w∈Rw\in\mathbb{R}, w+w=2ww+w=2w.

Indeed w+w=1⋅w+1⋅w=(1+1)w=2ww+w=1\cdot w+1\cdot w=(1+1)w=2w, by the multiplicative identity of R\mathbb{R}, distributivity, and the abbreviation 2=1+12=1+1 of the statement.

Claim 3 (clause 1). By Claim 1 and Addition Formulas for Sine and Cosine §cosine applied with a=ua=u and x=vx=v,

cos⁡(u−v)+cos⁡(u+v)=(cos⁡ucos⁡v+sin⁡usin⁡v)+(cos⁡ucos⁡v−sin⁡usin⁡v).\cos(u-v)+\cos(u+v)=\bigl(\cos u\cos v+\sin u\sin v\bigr)+\bigl(\cos u\cos v-\sin u\sin v\bigr).

Rearranging the four summands, the two terms sin⁡usin⁡v\sin u\sin v and −sin⁡usin⁡v-\sin u\sin v sum to 00, so the right-hand side is cos⁡ucos⁡v+cos⁡ucos⁡v\cos u\cos v+\cos u\cos v, which is 2cos⁡ucos⁡v2\cos u\cos v by Claim 2.

Claim 4 (clause 2). From the same two expansions,

cos⁡(u−v)−cos⁡(u+v)=(cos⁡ucos⁡v+sin⁡usin⁡v)−(cos⁡ucos⁡v−sin⁡usin⁡v),\cos(u-v)-\cos(u+v)=\bigl(\cos u\cos v+\sin u\sin v\bigr)-\bigl(\cos u\cos v-\sin u\sin v\bigr),

and now the two terms cos⁡ucos⁡v\cos u\cos v and −cos⁡ucos⁡v-\cos u\cos v sum to 00, leaving sin⁡usin⁡v+sin⁡usin⁡v\sin u\sin v+\sin u\sin v, which is 2sin⁡usin⁡v2\sin u\sin v by Claim 2.

Claim 5 (clause 3). By Addition Formulas for Sine and Cosine §sine applied with a=ua=u and x=vx=v, and by Claim 1,

sin⁡(u+v)+sin⁡(u−v)=(sin⁡ucos⁡v+cos⁡usin⁡v)+(sin⁡ucos⁡v−cos⁡usin⁡v).\sin(u+v)+\sin(u-v)=\bigl(\sin u\cos v+\cos u\sin v\bigr)+\bigl(\sin u\cos v-\cos u\sin v\bigr).

The two terms cos⁡usin⁡v\cos u\sin v and −cos⁡usin⁡v-\cos u\sin v sum to 00, leaving sin⁡ucos⁡v+sin⁡ucos⁡v\sin u\cos v+\sin u\cos v, which is 2sin⁡ucos⁡v2\sin u\cos v by Claim 2.

As u,v∈Ru,v\in\mathbb{R} were arbitrary, the three clauses hold for all u,v∈Ru,v\in\mathbb{R}.

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lemmaProduct-to-Sum Formulas for Sine and Cosinelem:trigonometric-product-formulas-2026a

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