Having trouble with graph of trig. function

I'm doing a graph sketching of

And I'm having trouble with two things:

1)

I get

Then, I checked on WolframAlpha and gives me an alternate form :

which is equivalent to what i got. But I have no idea how to go from the first form to the latter.

2)

When taking limits of f(x) as x-> infinity and -infinity, I understand:

as x->+-inifinty ; i get y=1 and y=-1 as the horizontal asymptotes.

Also:

as x->+-infinity; i get y=1 and y=0 as the horizontal asymptotes.

Then for lim of cos^4(x) as x->+-infinity I get the same thing, but when those two are added, the graph is between the lines y=.5 and y=1

Any help would be greatly appreciated.