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Hyperbolic functions are combinations of the exponential function analogous to trigonometric functions. The hyperbolic cosine and hyperbolic sine satisfy ${\cosh }^{2}x-{\sinh }^{2}x=1$ and parametrize a unit hyperbola.

Define a continuous map $F:S\to T$ by the standard embedded torus of revolution parametrizationIts image lies in $T$ becauseThe periodicity of the trigonometric functions makes $F(s,0)=F(s,1)$ and $F(0,t)=F(1,t)$, so $F$ is constant on every $\sim$-equivalence class. The universal property of the quotient topology therefore gives a unique continuous map

Every point of $T$ has cylindrical radius between $1$ and $3$, so its polar angle determines $s$ modulo one, while the pair $(\sqrt{x^2+y^2}-2,z)$ on the unit circle determines $t$ modulo one. Hence $\overline{F}$ is surjective, and two parameters have the same image exactly when they differ by the endpoint identifications defining $\sim$; thus it is injective.

The quotient $S/\sim$ is compact as the continuous image of the given compact space $S$ under the quotient map. The surface $T$ is Hausdorff because it has the subspace topology inherited from ${\mathbb{R}}^3$. The compact-to-Hausdorff continuous bijection theorem now shows that $\overline{F}$ is a homeomorphism. Therefore

Product-to-sum formulas turn products of trigonometric functions into sums at combined arguments. The identity $2\sin A\sin B=\cos (A-B)-\cos (A+B)$ is especially useful for verifying orthogonality of sine Fourier series modes on an interval.

A trigonometric function defined where $\cos x\ne 0$. It has real simple poles at $(n+1/2)\pi$, and $\tan ((n+1/2)\pi +\delta )=-\cot \delta$. This local pole identity generates the large roots near tangent poles.