I'm supposing this would be a UIL level problem

[sinkokuyou]

Calculate [unparseable or potentially dangerous latex formula] in terms of [unparseable or potentially dangerous latex formula] for some positive integers k and n and real numbers [unparseable or potentially dangerous latex formula].
I really have some trouble in understanding this problem...

[stupidityismygam]

This is probably above UIL level, but not above UIL conceptually.

From De Moivre's we have that
[unparseable or potentially dangerous latex formula]

From the binomial theorem we know that
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

So, comparing real parts,
[unparseable or potentially dangerous latex formula]

Since, [unparseable or potentially dangerous latex formula] we have that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula]

Thus, we acquire the result
[unparseable or potentially dangerous latex formula]

We can find [unparseable or potentially dangerous latex formula] as well in a similar fashion and see that
[unparseable or potentially dangerous latex formula]

This gives us
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

Seeing this answer suggests that there is a pretty nifty generalization

[AuSmith]

And just FYI, the sum [unparseable or potentially dangerous latex formula] is called a cosine series, which is a special type of Fourier series... used all the time in engineering.

[sinkokuyou]

umm, I understand how you did the problem, but I don't see how u used [unparseable or potentially dangerous latex formula] anywhere?

[stupidityismygam]

[AuSmith]

[stupidityismygam]