How to prove a filter has linear phase response?
Remember that there are four types of linear-phase FIR filters. The filter in your example is a type III filter: odd filter length and odd symmetry. The frequency response of such a system has the form

$$H(e^{j\omega})=A(\omega)je^{-j\omega(N-1)/2}\tag{1}$$

where $N$ is the filter length (number of taps), and $A(\omega)$ is real-valued and odd, i.e., $A(\omega)=-A(-\omega)$. Now it's up to you to show that your filter has the form given by $(1)$. Make use of 

$$\sin(x)=\frac{e^{jx}-e^{-jx}}{2j}\tag{2}$$

Note that because of the factor $j$ in $(1)$ and because $A(\omega)$ is odd, there's a phase jump of $\pi$ at $\omega=0$, so the phase is not strictly linear. Such a phase response is called generalized linear phase.