Check if 2 functions are linearly independent

Question

I have 2 functions:

$$f_1(x) = \cos(4x)$$ $$ f_2(x) = \cos(6x)$$ I used Wronskian and got

$W(x) = -6\cos(4x)\sin(6x)+4\sin(4x)\cos(6x)$

Now how can I tell if $W(x)$ equals $0$ (or not) with trigonometric identities?

Thanks :)

Answer 1

$$-6\cos(4x)\sin(6x) +4\sin(4x)\cos(6x) =0 \\ \implies 4\left(\sin(4x)\cos(6x) -\sin(6x)\cos(4x)\right) =2\cos(4x)\sin(6x) \\ \implies -4\sin(2x)=\sin(10x)+\sin(2x) \\ \implies \sin(10x) =-5\sin(2x)$$ This is not true for every $x$, for example take $x=\frac{\pi}{10}$.