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Exponential Identities

$e^{z}\equiv \sum _{n=0}^{\infty }{\frac {z^{n}}{n!}},$

$e^{z}=e^{\alpha _{0}}\left(\cos {\sqrt {\theta }}+(z-\alpha _{0}){\frac {\sin {\sqrt {\theta }}}{\sqrt {\theta }}}\right),$

where $\theta =Det(z-\alpha _{0})=(z-\alpha _{0})Adj(z-\alpha _{0})$ and $\alpha _{0}=Real(z)$

Trigonometry Identities

$\sin(z)\equiv \sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{2n+1}}{(2n+1)!}},$

$\sin(z)=\sin(\alpha _{0})\cosh {\sqrt {\theta }}+(z-\alpha _{0})\cos(\alpha _{0}){\frac {\sinh {\sqrt {\theta }}}{\sqrt {\theta }}},$

$\cos(z)\equiv \sum _{n=0}^{\infty }(-1)^{n}{\frac {z^{2n}}{(2n)!}},$

$\cos(z)=\cos(\alpha _{0})\cosh {\sqrt {\theta }}-(z-\alpha _{0})\sin(\alpha _{0}){\frac {\sinh {\sqrt {\theta }}}{\sqrt {\theta }}},$

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.

Hyperbolic Trigonometry Identities

$\sinh(z)\equiv \sum _{n=0}^{\infty }{\frac {z^{2n+1}}{(2n+1)!}},$

$\sinh(z)=\sinh(\alpha _{0})\cos {\sqrt {\theta }}+(z-\alpha _{0})\cosh(\alpha _{0}){\frac {\sin {\sqrt {\theta }}}{\sqrt {\theta }}},$

$\cosh(z)\equiv \sum _{n=0}^{\infty }{\frac {z^{2n}}{(2n)!}},$

$\cosh(z)=\cosh(\alpha _{0})\cos {\sqrt {\theta }}+(z-\alpha _{0})\sinh(\alpha _{0}){\frac {\sin {\sqrt {\theta }}}{\sqrt {\theta }}},$