Taking your second equation an setting $t_1,t_2\to t$, we get $$E(k)=\pm t \sqrt2 \sqrt{1+\cos(ka)}=\pm 2 t \cos\left(\frac{ka}2\right)$$
where I used $1+\cos x = 2 \cos x/2$. The prefactor $2t$ comes because your Hamiltonian is degenerate (try to put $t_1,t_2\to t$ in the second Hamiltonian). The $k/2$ comes because $N$ is different by a factor of 2.