As #sinx+cosx=sqrt2sin(x+pi/4)#, in the interval #(-pi,-pi/2)uu(0,pi/2)# #arcsin(sinx+cosx)# is not define i.e. #x# has valid values only in the interval #[-pi/2,0]uu[pi/2,pi]#
graph{sinx+cosx [-5, 5, -2.5, 2.5]}
For valid domain of #[-pi/2,0]uu[pi/2,pi]#, #cos{arcsin(sinx+cosx)}# takes values from #0# to #1# and #0# to #-1# and hence range of #f(x)=1/cos{arcsin(sinx+cosx)}#, is #(-oo,-1]uu[1,oo)#
Observe that it is also true for the range of #secx# hence while range is not affected, domain of #x# is limited as for some values of #x#, #sinx+cosx# takes values beyond #[-1,1]#.
