How can we sum up $\sin^m$ and $\cos^m$ series when the angles are in arithmetic progression?

Question

How can we sum up $\sin^m$ and $\cos^m$ series when the angles are in arithmetic progression?

1. Does an identity exist similar to 1.1 and 1.2 for $m>1$?, and
2. Does an approximate or estimate exist?

For example here is the sum of $\cos$ series when $m=1$:

$$\sum_{k=0}^{n-1}\cos^m (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl( \frac{ 2 a + (n-1)\cdot d}{2}\biggr) \,\,\,(1.1)$$

There is a slight difference in case of $\sin$, which is: $$\sum_{k=0}^{n-1}\sin^m (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \sin\biggl( \frac{2 a + (n-1)\cdot d}{2}\biggr) \,\,\,(1.2)$$

As you can see, in the case of $m=1$ an exact identity exists that is simple enough.

Answer 1

Once you have a formula for $\cos^m$ you get a formula for $\sin^m$ using $\sin(x)=\cos(\pi/2-x).$ An arithmetic sequence subtracted from $\pi/2$ is still an arithmetic series.

So we only need to think about $\cos^m.$

For $m=2,$ you can use that $$\cos^2(x)=\frac12(\cos(2x)+1).\tag{2}$$

For $m=3,$ you can use: $$\cos^3(x)=\frac14(\cos(3x)+3\cos(x))\tag{3}.$$

We can use these formulae to get formula for $m=2,3.$

For example, for $m=2$ you get:

$$\sum\cos^2=\frac{n}2+\frac12\frac{\sin nd}{\sin d}\cos(2a+(n-1)d)$$

The formula for $\sin^2$ is easy to derive from this:

$$\sum\sin^2=\frac{n}2-\frac12\frac{\sin nd}{\sin d}\cos(2a+(n-1)d)$$

For any $m,$ there is such a formula like (2) and (3), namely for $m$ odd:

$$\cos^m(x)=\frac1{2^{m-1}}\left(\sum_{k=0}^{\lfloor m/2\ \rfloor}\binom{m}{k}\cos(m-2k)x\right)$$

When $m$ is even, you have to adjust the the constant term:

$$\cos^m(x)=\frac1{2^{m-1}}\left(\frac12\binom{m}{m/2}+\sum_{k=0}^{ m/2-1}\binom{m}{k}\cos(m-2k)x\right)$$

Answer 2

The powers of the sine and cosine can be "linearized" in terms of sine and cosine of multiples of the argument, by

$$\sin^m x=\left(\frac{e^{ix}-e^{-ix}}{2i}\right)^m$$ and similar for the cosine. After expansion, sines and cosines reappear.

Hence the formulas you are after are found as linear combinations of the formulas for $m=1$, where the coefficients are binomial coefficients over powers of $2$.