Having trouble with graph of trig. function

I'm doing a graph sketching of $\displaystyle f(x)=sin^4(x) + cos^4(x)$

And I'm having trouble with two things:

1)

I get $\displaystyle f''(x)=-4sin^4(x) -4cos^4(x) + 24sin^2(x)cos^2(x)$

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

$\displaystyle f''(x)=-4cos(4x)$ 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:

$\displaystyle lim (sin(x)) $as x->+-inifinty ; i get y=1 and y=-1 as the horizontal asymptotes.

Also:

$\displaystyle lim (sin(x))^4$ 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.