The other tutor's answer is wrong.

You don't need to simplify it.

Use the theorem that if f is continuous in a neighborhood of a,
and is not constant on any subinterval of that interval
and lim(x/y) as x->a exists,

lim [f(x)/f(y)] = lim(x/y)
x->a              x->a

then since cos(0) = 1,
                                               _                 _
lim  log[log(1-x^2)]      lim   1-x^2     lim |    1         x^2  |
     ----------------- =       ------- =      | ------- - --------| = 1-0 = 1
x->0  log[log(cos(x)]     x->0  cos(x)    x->0|_ cos(x)    cos(x)_|

Edwin