Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent. Prove that the set $ \{\sin(x),\cos(x),\sin(2x),\cos(2x)\}$ is linearly independent.

I have tried plugging in values for $x$ but where does this lead? I know what the Wronksian is and it would give an almost trivial solution in this case, but I am not allowed to use that.

Let $a,b,c,d\in\mathbb{R}$ such that $x\mapsto a\sin x + b\cos x+ c\sin 2x + d\cos 2x$ is the $0$ function. You have to show that one must have $a=b=c=d=0$.
For $x=0$, we get $b+d=0$. For $x=\pi$, $-b+d=0$. Combining the two, $b=d=0$.
Now, you have $a\sin x + c\sin 2x = 0$ for all $x$. Can you use similar techniques (or new ones -- differentiating and then using the same technique, for instance) to show that $a=c=0$ as well? (even without using differentation, just considering $x=\frac{\pi}{2}$ may help).
