Finding Smallest Integer for Integer Expression with Trigonometric Functions

[ehrenfest]

[SOLVED] Aime 2008 Ii 8

Homework Statement

Let $a = \pi/2008$. Find the smallest positive integer n such that
$$2[\cos(a)\sin(a)+\cos(4a)\sin(2a)+\cos(9a)\sin(3a)+\cdots+\cos(n^2a)\sin(na)]$$
is an integer.

Homework Equations

$$\cos(a+b) = \cos a \cos b- \sin a \sin b$$

$$\sin (a+b) = \sin a \cos b + \sin b \cos a$$

The Attempt at a Solution

Can someone give me a hint please? This should only require high school math. I am not sure if the identities above are useful here or if there is a totally different method needed.

 

[Tedjn]

It's an AIME trig problem, which often means you have to play around with it and hope things end up canceling. You have a product which is difficult to sum, so try changing the product into a sum using the sum to product identities:

http://www.mathwords.com/s/sum_to_product_identities.htm

 

[ehrenfest]

very nice