Q:

Finding coordinate vector of $f(t) = \cos(t+a)$ relative to the basis $\{\sin(t+2\pi), \cos(t+2\pi)\}$

Ok so, 
$$\cos(t+a) = \cos(t)\cos(a) - \sin(t)\sin(a) $$
$$= \cos(t+2\pi) \cos(a) - \sin(t+2\pi)\sin(a) $$ 
Would this mean that the coordinate vector is $(1,-1)$?  Or should i find the coordinates in a different form? 

A:

You basically need to find the coefficients of the linear combination $sin(t+2\pi) + cos(t+2\pi) = cos(t+a)$.
Your trigonometric identity is a good path. In this case, I believe it means that the coordinate vector is $cos(a)$, $sin(a)$ as these are the coefficients in front of the basis components. 

