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

In the vector space of $f:\mathbb R \to \mathbb R$, how do I prove that functions $\sin(x)$ and $\cos(x)$ are linearly independent. By def., two elements of a vector space are linearly independent if $0 = a\cos(x) + b\sin(x)$ implies that $a=b=0$, but how can I formalize that? Giving $x$ different values? Thanks in advance.

Answer

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