Fungrim entry: 3fb3ca

\sin\!\left(z\right) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}

Assumptions:

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}

Alternative assumptions:

z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}

TeX:

\sin\!\left(z\right) = \frac{2 \tan\!\left(\frac{z}{2}\right)}{\tan^{2}\!\left(\frac{z}{2}\right) + 1}

z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}

z \in \mathbb{C}[[x]] \,\mathbin{\operatorname{and}}\, z \notin \left\{ \left(2 n + 1\right) \pi : n \in \mathbb{Z} \right\}

Definitions:

Fungrim symbol	Notation	Short description
`Sin`	$\sin\!\left(z\right)$	Sine
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ConstPi`	$\pi$	The constant pi (3.14...)
`ZZ`	$\mathbb{Z}$	Integers
`FormalPowerSeries`	$K[[x]]$	Formal power series

Source code for this entry:

Entry(ID("3fb3ca"),
    Formula(Equal(Sin(z), Div(Mul(2, Tan(Div(z, 2))), Add(Pow(Tan(Div(z, 2)), 2), 1)))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotElement(z, SetBuilder(Mul(Add(Mul(2, n), 1), ConstPi), n, Element(n, ZZ)))), And(Element(z, FormalPowerSeries(CC, x)), NotElement(z, SetBuilder(Mul(Add(Mul(2, n), 1), ConstPi), n, Element(n, ZZ))))))

Topics using this entry

Sine