Applications of higher powers of trigonometric functions I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers.
An example is one that I have been using in my own studies:
$\cos^4 \theta$
I am interested in further examples of applications of these types of functions, particularly if they are trigonometric identities.
 A: 1) Two applications I can think of relate to Wallis integral : Considering the sequence $(I_{n})_{n \geq 0}$ defined by
$$ I_{n} = \int_{0}^{\frac{\pi}{2}} \sin^{n}(x) \: dx = \int_{0}^{\frac{\pi}{2}} \cos^{n}(x) \: dx $$
one can prove the Stirling formula (see http://en.wikipedia.org/wiki/Wallis%27_integrals) :
$$ n! \sim \left( \frac{n}{e} \right)^{n} \sqrt{2n\pi} $$
or the Wallis product (see http://en.wikipedia.org/wiki/Wallis_product#cite_note-2) :
$$ \prod_{n=1}^{+\infty} \frac{2n}{2n-1} \frac{2n}{2n+1} = \frac{\pi}{2} $$
-- Edit : another application of Wallis integral :
Let $R >0$ and $n \in \mathbb{N}^{\ast}$. Let $B = \lbrace x=(x_{1},\ldots,x_{n}) \in \mathbb{R}^{n}, \, \Vert x \Vert_{2} \leq R \rbrace$ be the unit ball of $\mathbb{R}^{n}$ and let $V_{n}$ be its volume. Using Wallis integral, one can prove that :
$$ V_{2n}=\frac{\pi^{n}}{n!} R^{2n} $$
and
$$ V_{2n+1} = 2^{2n+1} \frac{n!}{(2n+1)!} \pi^{n} R^{2n+1} $$
and
$$ \lim \limits_{n \rightarrow +\infty} V_{n} = 0 $$
2) Using Euler identities ($\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and $\sin(x) = \frac{e^{ix}-e^{-ix}}{2i}$) and the binomial theorem (see http://en.wikipedia.org/wiki/Binomial_theorem), one can prove the following identities :
$$ \cos^{2n}(x) = \frac{1}{4^{n}} \left( \begin{pmatrix} 2n \\ n \end{pmatrix} + 2 \sum_{k=0}^{n-1} \begin{pmatrix} 2n \\ k \end{pmatrix} \cos \left( 2(n-k)x \right) \right) $$
and
$$ \cos^{2n+1}(x) = \frac{1}{4^{n}} \sum_{k=0}^{n} \begin{pmatrix} 2n+1 \\ n-k \end{pmatrix} \cos((2k+1)x) $$
(and there are similar identities for $\sin^{2n}(x)$ and $\sin^{2n+1}(x)$. These identities allow you to integrate powers of $\cos$ and $\sin$.
