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.

suppose $$a \cos x + b \sin x + c \cos 2x + d \sin 2x = 0$$ for all $x.$ putting $x = 0$ gives $$a + c = 0 \tag 1 $$
put $x = \pi$ gives $$ -a +c= 0 \tag 2$$
$(1)$ and $(2)$ shows $a = 0, c =0.$
put $x = \pi/2$ gives $ b = 0 $ now we are left with $d \sin 2x = 0$ which implies $d = 0.$
so we have shown that $a \cos x + b \sin x + c \cos 2x + d \sin 2x = 0$ implies $a=b=c=d=0$ which means $\{\cos x,\sin x,  \cos 2x,\sin 2x  \}$ is linear independent.
