10. Product-to-sum and sum-to-product identities

The product-to-sum identities or prosthaphaeresis formulas can be proven by expanding their right-hand sides using the angle addition theorems. See amplitude modulation for an application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae.

Product-to-sum
${\displaystyle 2\cos \theta \cos \varphi ={\cos(\theta -\varphi )+\cos(\theta +\varphi )}}$
${\displaystyle 2\sin \theta \sin \varphi ={\cos(\theta -\varphi )-\cos(\theta +\varphi )}}$
${\displaystyle 2\sin \theta \cos \varphi ={\sin(\theta +\varphi )+\sin(\theta -\varphi )}}$
${\displaystyle 2\cos \theta \sin \varphi ={\sin(\theta +\varphi )-\sin(\theta -\varphi )}}$
${\displaystyle \tan \theta \tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}$
${\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where }}S=\{1,-1\}^{n}\end{aligned}}}$

Sum-to-product
${\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}$
${\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}$
${\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\theta +\varphi \over 2}\right)\sin \left({\theta -\varphi \over 2}\right)}$

Other related identities

• If x + y + z = π (half circle), then

  ${\displaystyle \sin(2x)+\sin(2y)+\sin(2z)=4\sin x\sin y\sin z.\,}$

• Triple tangent identity: If x + y + z = π (half circle), then

  ${\displaystyle \tan x+\tan y+\tan z=\tan x\tan y\tan z.\,}$

  In particular, the formula holds when x, y, and z are the three angles of any triangle.

  (If any of x, y, z is a right angle, one should take both sides to be ∞. This is neither +∞ nor −∞; for present purposes it makes sense to add just one point at infinity to the real line, that is approached by tan θ as tan θ either increases through positive values or decreases through negative values. This is a one-point compactification of the real line.)

• Triple cotangent identity: If x + y + z = π/2 (right angle or quarter circle), then

  ${\displaystyle \cot x+\cot y+\cot z=\cot x\cot y\cot z.\,}$

Hermite's cotangent identity

Charles Hermite demonstrated the following identity. Suppose a₁, ..., a_n are complex numbers, no two of which differ by an integer multiple of π. Let

${\displaystyle A_{n,k}=\prod _{\begin{smallmatrix}1\leq j\leq n\\j\neq k\end{smallmatrix}}\cot(a_{k}-a_{j})}$

(in particular, A_1,1, being an empty product, is 1). Then

${\displaystyle \cot(z-a_{1})\cdots \cot(z-a_{n})=\cos {\frac {n\pi }{2}}+\sum _{k=1}^{n}A_{n,k}\cot(z-a_{k}).}$

The simplest non-trivial example is the case n = 2:

${\displaystyle \cot(z-a_{1})\cot(z-a_{2})=-1+\cot(a_{1}-a_{2})\cot(z-a_{1})+\cot(a_{2}-a_{1})\cot(z-a_{2}).}$

Ptolemy's theorem

Ptolemy's theorem can be expressed in the language of modern trigonometry as:

If w + x + y + z = π, then:

${\displaystyle {\begin{aligned}\sin(w+x)\sin(x+y)&=\sin(x+y)\sin(y+z)&{\text{(trivial)}}\\&=\sin(y+z)\sin(z+w)&{\text{(trivial)}}\\&=\sin(z+w)\sin(w+x)&{\text{(trivial)}}\\&=\sin w\sin y+\sin x\sin z.&{\text{(significant)}}\end{aligned}}}$

(The first three equalities are trivial rearrangements; the fourth is the substance of this identity.)