Hm, I've followed your steps but I'm struggling to grasp how you're able to replace cos(x) with u and find the maximum of f(x) = (cos x - 1/2)^2 - 1/16 based on f(u) = (u-1/2)^2 - 1/16 for -1<= u <=1
We know that no matter what values of
we put into
, the smallest number we can get out is
and the largest number out is
. That is,
. So we can replace the
in our function with another variable that can take these same values, so that you're effectively getting the same values of
back out. If you plot
for those values you'll get the same values out if you were to plot
for all values of
. (see how the
values are the same
https://www.desmos.com/calculator/zmjvisslih). So you can then maximise your new function to maximise the old one. The only change will be that you'll be putting a different value into that function to get the maximum, so you'd find it at a different value of
than
, but if you were asked to find this value of
, you know how the two are related (
), and so can work it back.
Hope that helps