So, here are two functions $x_1(t)=\cos(2t)-1 \text{ and } x_2(t)=\sin^2(t), I = R$. $C_1x_1(t)+C_2x_2(t)=0$
Here I am stuck, because I try to find the values of $t$ such that I can express $C_1 \text{ or } C_2 $ and show that they must be equal to zero. But in this case, no matter the $t$ I choose I get all expression equal to 0, so I cannot conclude that $C_1 \text{ and} C_2 \text{ must be equal to } 0 \text{ so that the equation } C_1x_1(t)+C_2x_2(t)=0 \text{ is valid}.$ I think may be I can use Wronskian determinant as it tells whether two functions are linearly dependant. But the determinant was very complicated, so I think there must be some other solution.