The set is:

{1, cos(2x) 3*(sin(x))^2}

I figure that:

C1(1 +C2(cos(2x) + C3*(3*sin(x))^2 =0

That gives us:

C1(1) + C2(Cos(0) + C3(3(sin(0))^2 = 0

Thus C1 + C2 + 0 = 0

So:

C1 + C2 = 0

How do I find C3?

If my process is correct...Linear dependence exists if either C1 , C2 or C3 are non zero...