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.

If Cos(x) and sin(x) are linearly independent, then this implies.
acos(x)+bsin(x)=0 if and only if (for all values of x) a=b=0.
But!! For x=pi/4 rad; cos(pi/4)=sin(pi/4)= sqrt(2)/2 and for a=1 and b=-1
we get: (1)*sqrt(2)/2 +(-1)*sqrt(2)/2 which is sqrt(2)/2-sqrt(2)/2=0.
Therefore: cos(x) and sin(x) cannot be linearly independent.
