Infinite series of trigonometric terms

[DeathbyGreen]

I'm trying to make an approximation to a series I'm generating; the series is constructed as follows:
Term 1:
$\left[\frac{cos(x/2)}{cos(y/2)}\right]$

Term 2:
$\left[\frac{cos(x/2)}{cos(y/2)}-\frac{sin(x/2)}{sin(y/2)}\right]$

I'm not sure yet if the series repeats itself or forms a pattern; but if it continues to add terms proportional to sine and cosine half angle fractions, are there any series I could use to express an infinite number of these types of terms as an exact form? I've looked at a Fourier series but I'm not sure it would work. Thank you!

 

[mfb]

There is an infinite set of series that (a) have these two initial terms, (b) have a relatively compact way to write down their terms and (c) have an exact and compact way to write down their limit. I'm not sure if you ask for (b) or (c), but both together are possible as well.

What is your actual problem that you want to solve?

 

[DeathbyGreen]

Thanks for the quick reply! The problem I'm trying to solve is complicated, but I'm just looking for mathematical identities. Which set of series are you referring to? I'm looking for (c), the limit as the series goes to infinity, but a nice compact way to write down the terms (b) would be nice too!

 

[mfb]

You don't constrain your series at all. All following terms could be zero. That makes it trivial to evaluate the limit.