the maximum value of cosx = 1 at 2npi and its minimum value is -1 at (2n+1)pi.
so we put this cos(cosx).
we get the maximum value to be cos(cos0). we don't consider cos(cos180) = cos1 for the maximum value. it will be lesser than 1.
yeah for the minimum value we take cos(cos90) or cos(cos180) = cos1. cos(-x) = cosx.
as x increases from 0 to 90 cosine decreases.
the range is form the maximum value to the minimum value. :-)
Input of cos(cosx) is cosx which lies between -1 and 1.
Since cos(-x) = cosx it is sufficient to solve for cosx varying from 0 to 1.
from 0 to 1 cosx is decreasing function and hence its maximum will occur at cosx=0 and minimum at cosx=1
so cos(cosx) will vary from cos(1) to cos(0) i.e. (cos1 , 1)