  We study the hyperbolic cosine and sine laws in the extended hyperbolic space
which contains hyperbolic space as a subset and is an analytic continuation of
the hyperbolic space. And we also study the spherical cosine and sine laws in
the extended de Sitter space which contains de Sitter Space $S^n_1$ as a subset
and is also an analytic continuation of de Sitter space. In fact, the extended
hyperbolic space and extended de Sitter space are the same space only differ by
-1 multiple in the metric. Hence these two extended spaces clearly show and
apparently explain that why many corresponding formulas in hyperbolic and
spherical space are very similar each other. From these extended trigonometry
laws, we can give a coherent and geometrically simple explanation for the
various relations between the lengths and angles of hyperbolic polygons and
relations on de Sitter polygons which lie on $S^2_1$.
