p = (((Cos[t] + Sin[t]))^10)

q = TrigReduce[p]

Expand[q]

(1/(2  π)) (∫_0^(2  π))p \[DifferentialD]t

Table[(1/π) (∫_0^(2  π))p Cos[
              k t] \[DifferentialD]t, {k, 1, 10}]

Table[(1/π) (∫_0^(2  π))p Sin[
              k t] \[DifferentialD]t, {k, 1, 10}]