令x=2kπ+t,t∈[0,2π)
當x=2kπ,2kπ+π/2時,arccos(cosx)=arcsin(sinx)
當x=2kπ+π,2kπ+3π/2時,arccos(cosx)>arcsin(sinx)
(1)、當t∈(0,π/2)時,
arccos(cosx)=t∈(0,π/2)
arcsin(sinx)=t∈(0,π/2)
arccos(cosx)=arcsin(sinx)
(2)、當t∈(π/2,π)時,
arccos(cosx)=t∈(π/2,π)
arcsin(sinx)=π-t∈(0,π/2)
arccos(cosx)>arcsin(sinx)
(3)、當t∈(π,3π/2)時
arccos(cosx)=2π-t∈(π/2,π)
arcsin(sinx)=π-t∈(-π/2,0)
arccos(cosx)>arcsin(sinx)
(4)、當t∈(3π/2,2π)時,
arccos(cosx)=2π-t∈(0,π/2)
arcsin(sinx)=t-2π∈(-π/2,0)
arccos(cosx)>arcsin(sinx)
綜上所訴,
x∈(2kπ+π/2,2kπ+2π)