$\sin x$ and $\cos x$ - linearly independent There is a lot of posts about that subject, but I'm not sure.
Task: 
Check if 
$$f(x)=\sin(x)  $$
and 
$$g(x)=\cos(x)  $$
are linearly independent in the space of functions.
That's what I have done: 
$a \sin(x) + b \cos(x) = 0$, 
so $a=0$ and $b=0$.
for $x=0  $, 
$a \sin(0) + b \cos(0) = 0$, then 
$a\cdot  0 + b\cdot 1 = 0$.
for $x=\frac{\pi}{2}$
$a \sin(\frac{\pi}{2}) + b \cos(\frac{\pi}{2}) = 0$, then 
$b \cdot 1 + a \cdot  0 = 0$
what am I supposed to write then? :)

The Wronskian of $\sin$ and $\cos$ is not zero:
$$W(\sin, \cos)(x) = \begin{vmatrix} \sin x & \cos x \\ \cos x & -\sin x\end{vmatrix} = -\sin^2 x - \cos^2 x = -1 \ne 0$$
Therefore $\{\sin, \cos\}$ is linearly independent.
