Fungrim entry: 54f420

\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

TeX:

\sin^{2 n}\!\left(z\right) = \frac{1}{{4}^{n}} {2 n \choose n} + \frac{2}{{4}^{n}} \sum_{k=0}^{n - 1} {\left(-1\right)}^{n + k} {2 n \choose k} \cos\!\left(2 \left(n - k\right) z\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Sin`	$\sin(z)$	Sine
`Binomial`	${n \choose k}$	Binomial coefficient
`Sum`	$\sum_{n} f(n)$	Sum
`Cos`	$\cos(z)$	Cosine
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("54f420"),
    Formula(Equal(Pow(Sin(z), Mul(2, n)), Add(Mul(Div(1, Pow(4, n)), Binomial(Mul(2, n), n)), Mul(Div(2, Pow(4, n)), Sum(Mul(Mul(Pow(-1, Add(n, k)), Binomial(Mul(2, n), k)), Cos(Mul(Mul(2, Sub(n, k)), z))), For(k, 0, Sub(n, 1))))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(0)))))

Topics using this entry

Sine