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.

Does this work?
Assume that $\sin(x)$ and $\cos(x)$ are linearly dependent. Then one must be a scalar multiple of the other, that is
$$\sin(x) = a * \cos(x)$$
But for $x=\pi/2$, we have
$$1 = a * 0$$
which is impossible. Thus $\sin(x)$ and $\cos(x)$ are linearly independent.
I have a feeling that there is some error in my reasoning here, please point it out if you see one :)
