Let #f:RR -> A#, #f(x)=cos(sinx)#.
There are multiple ways to go about this. We can use the fact that #f# is a composed function or do it directly. We'll use the second approach.
We know that the range of #sinx# and #cosx# is #[-1,1]#. Hence, The range of #cos(sinx)# will be located in the interval #[-1,1]#. However, the exact range, denoted #A# in this answer, is going to be:
#A=[cos(x_1),cos(x_2)]#
Where #cos(x_1)# is the minimal cosine value in the interval #[-1,1]# and #cos(x_2)# is the maximum value. As #cos(x)# is periodic, so will #cos(sinx)#.
On the interval #[-1,0]# the cosine function is monotonic increasing, hence the minimal value of #cosx# is going to be when #x=-1#. Similarly, as the cosine is monotic decreasing on #[0,1]#, this means #x=0# is the max value and it also means that #x=1# is also the minimal value.
#:. A = [cos(-1), cos0]=[cos1,cos0]=[cos1,1]#
Thus, the range is
#color(red)(A=[cos1, 1]#
