expansion of $\cos^k(\theta)$

Question

Does any body know a expansion of : $\cos^k(\theta)$ in function of $\cos$ and/or $\sin$ but without power? For example : $\cos^2(\theta)=\frac{1}{2}(\cos(2\theta)+1)$, but i would want a generalization to the power $k$. I searched in Abramowitz and Stegun but didn't find anything !

Answer 1

$$ \cos^k \theta = \left( \frac{ e^{i\theta} + e^{-i\theta} }{2} \right)^k = \frac{1}{2^k} \sum_{n=0}^k \binom{k}{n} (e^{i\theta} )^n (e^{-i\theta})^{k-n} = \frac{1}{2^{k+1} } \sum_{n=0}^k 2\binom{k}{n} e^{i(2n-k)\theta} $$

Since $ \binom{k}{n} = \binom{k}{k-n} $ we split up the sum and reverse the index of summation in the second term as follows:

$$ = \frac{1}{2^{k+1} } \left( \sum_{n=0}^k \binom{k}{n} e^{i(2n-k)\theta} + \sum_{n=0}^k \binom{k}{k-n} e^{i(2n-k)\theta}\right) = \frac{1}{2^{k+1} } \left( \sum_{n=0}^k \binom{k}{n} e^{i(2n-k)\theta} + \sum_{n=0}^k \binom{k}{n} e^{i(k-2n)\theta}\right) $$

$$ = \frac{1}{ 2^{k} } \sum_{n=0}^k \binom{k}{n} \left( \frac{ e^{i(2n-k)\theta} + e^{-i(2n-k)\theta} }{2} \right) =\frac{1}{ 2^{k} } \sum_{n=0}^k \binom{k}{n} \cos ( (2n-k)\theta). $$

Answer 2

I just wish to contribute a "quicker" development of Ragib Zaman's derivation. Just as he had shown, $$ \cos^k \theta = \left( \frac{ e^{i\theta} + e^{-i\theta} }{2} \right)^k = \frac{1}{2^k} \sum_{n=0}^k \binom{k}{n} (e^{i\theta} )^n (e^{-i\theta})^{k-n} = \frac{1}{2^k } \sum_{n=0}^k \binom{k}{n} e^{i(2n-k)\theta} $$ Now, assuming that $\theta$ is real, we know that $\cos \theta \in \mathbb{R}$, so both sides of the above equation must be real. Thus, we may take the real part on both sides with no change on the left hand side, i.e. $$ \cos^k \theta = \text{Re} \left [ \frac{1}{2^k } \sum_{n=0}^k \binom{k}{n} e^{i(2n-k)\theta} \right ] = \frac{1}{2^k } \sum_{n=0}^k \binom{k}{n}\;\, \text{Re} \left [ e^{i(2n-k)\theta} \right ], $$ whereby $$ \cos^k \theta = \frac{1}{2^k } \sum_{n=0}^k \binom{k}{n}\;\, \cos \left[ (2n - k) \theta \right ] $$

Answer 3

The example you give can be generalized using trigonometric identities. For odd $k$ we get $$\cos^k(\theta)=2^{1-k}\sum\limits_{n=0}^{\frac{k-1}{2}}\binom{k}{n}\cos((k-2n)\theta)$$ while for even $k$ the formula becomes $$\cos^k(\theta)=2^{-k}\binom{k}{\frac{k}{2}}+2^{1-k}\sum\limits_{n=0}^{\frac{k}{2}-1}\binom{k}{n}\cos((k-2n)\theta).$$

Answer 4

You can repeatedly replace powers of the cosine by a combination of smaller powers and $\cos(n\theta)$ using Chebyshev polynomials of the first kind. I'm not sure the resulting coefficients have any conventional name, though.

Alternatively, express $\cos x$ as a Fourier series and use the second convolution theorem to raise it to the $k$th power.