Linear independence of $\sin(x)$ and $\cos(x)$

Hint: If $a\cos(x)+b\sin(x)=0$ for all $x\in\mathbb{R}$ then it is especially true for $x=0,\frac{\pi}{2}$

There are easier ways - eg take special values for $x$. The following technique is a sledgehammer in this case, but a useful one to have around.

Suppose you have $a$ and $b$ as required. Let $r=\sqrt{a^2+b^2}$ and $\phi = \arctan {\frac a b}$ (take $\phi=\frac {\pi} 2$ if $b=0$). Then we have: $$a \cos (x)+b\sin(x)=r\sin(\phi)\cos(x)+r\cos(\phi)\sin(x)=r\sin(x+\phi)$$

The last form is identically zero only if $r=0$, which immediately implies $a=b=0$ from the definition of $r$.