Prove that the functions cos3x, sin3x, cos4x, sin4x are linearly independent in ODEs

Question

i tries to solve using the Wronskian test for independence, but the answer is weird please help Prove that the functions cos3x, sin3x, cos4x, sin4x are linearly independent.

Answer 1

If $a\cos 3x+b\sin 3x+c\cos 4x+d\sin 4x=0$ for all $x$, the case $x=0$ gives $a+c=0$ while $x=\pi$ gives $c-a=0$, so $a=c=0$. The case $x=\pi/2$ gives $-b=0$ so $b=0$, while the case $x=\pi/3$ gives $d\sin\frac{4\pi}{3}=0$ so $d=0$.