Expressing $\cos(x)^6$ as a linear combination of $\cos(kx)$'s Let $$(\cos^6(x)) = m\cos(6x)+n\cos(5x)+o\cos(4x)+p\cos(3x)+q\cos(2x)+r\cos(x)+a.$$ 
What is the value of $a$?

If:
$$\cos^6(x)=\sum_{k=1}^6 a_k \cos(kx)+a_0$$ 
Then we have:
$$\cos^6(0)+\cos^6(\pi)=2=a_2+ a_4+a_6+2a_0$$ 
$$\cos^6(3\pi/4)+\cos^6(\pi/4)=1/4=-2a_4+2a_0$$
$$\cos^6(3\pi/2)+\cos^6(\pi/2)=0=-2a_2+2a_4-2a_6+2a_0$$
Add them all up to find $a_0$.
