a trigonometric identity

A trigonometric identity
For any α(1),..,α(m),
sin(α(1)+..+α(m)) = Σ_rP(m,r)S(r)sin(α(r)),
where
P(m,m) = 1 and P(m,r) = cos(α(r+1))..cos(α(m)) for r < m, and
S(1) = 1 and S(r) = cos(α(1)+..+α(r-1)) for r > 1.

The proof is a straight-forward induction on m, using the formula
sin(a+b) = sin(a)cos(b)+cos(a)sin(b).

There is a hyperbolic analogue. The formulae for sin(a+b) and sinh(a+b)
are essentially the same, so the same proof yields

A hyperbolic trigonometric identity
For any α(1),..,α(m),
sinh(α(1)+..+α(m)) = Σ_rPh(m,r)Sh(r)sinhα(r)),
where
Ph(m,m) = 1 and Ph(m,r) = cosh(α(r+1))..cosh(α(m)) for r < m, and
Sh(1) = 1 and Sh(r) = cosh(α(1)+..+α(r-1)) for r > 1.

hyperbolic geometry