If y= Acoskt + Bsinkt where A, B, and k are constants, show that y"+k^2y=0

i did this

y"= -Aksinkt + Bkcoskt

=y" + k^2y
=-Aksinkt + Bkcoskt + k^2(Acoskt + Bsinkt)
=-Aksinkt + Bkcoskt + Ak^2coskt + Bk^2sinkt

is this right, how do i do the rest? please and thank you

1. Find y''
2. plug in function of y'' and y in the function y"+k^2y=0.
(-Aksinkt + Bkcoskt) + k^(2(Acoskt + Bsinkt))=0, assuming you did y'' correctly
3. if everything in the left cancels out, then you did it right