Is this a valid sinusoid?
The 2nd graph is not exactly periodic.

In the second graph, magnitude of X[-5] must = x[7] (this is because the change of n from the local minima is similar), however doesn't.
Similarly X[-7]  must = X[5] if their distance from the local minima is equivalent.

The only time this second graph may be valid is if there's a function that is modulating the amplitude based on the input n  like f(n)cos(n), however here there's only frequency modulation  cos(\$\alpha\$ n)

 Correction 

Second graph is actually valid

Although it doesn't seem periodic due to limited samples. It is symmetric from the origin, and if you calculate it out, the period is at 12.5, which is why there's a X[12] is = x[13] = 1 (roughly).