Is it possible to apply trigonometric conversion on multiple sum of cosine functions ?
I have a sum like this:
$$\cos(\pi a_1) + \cos(\pi a_2) + \cos(\pi a_3) + ... + \cos(\pi a_{10})$$
The coefficient values $a_k$ can be integer or floating point value. Generally trigonometric conversions are developed on sum of two cosine functions.
Yes, a sum of two cosines can be expressed as a product of two cosines, but a sum of more than two cosines is, in general, as simple as possible. Of course, it is possible to has a product $\ n\ $ cosines and expand it into the sum of $\ 2^{n-1}\ $ cosines, but these cosines have a special relationship with each other.
To get a good idea of this, expand the product $\ (x+\frac1x)(y+\frac1y)(z+\frac1z)\ $ as $$ (x y z + \frac1{x y z}) + (\frac{y z}x + \frac{x}{y z}) + (\frac{x z}y + \frac{y}{x z}) + (\frac{x y}z + \frac{z}{x y}). $$ This is easily generalized and applies to cosines since one definition of cosine is $\ \cos(x) := (e^{i x} + 1/e^{i x})/2.$ Perhaps it is easier to understand this as the trig identity $$ cos(x)cos(y) = (\cos(x+y)+\cos(x-y))/2.$$