Question

cos(X+pi/6) + sin(x-pi/3) = 0

I am working with one side but I am not sure if I can subtract at all here.

Answer 1

To see if it is an identity, pick any value of x
e.g. x = 78°
then x+π/6 ---> 78+30 = 108°
and x-π/3 ---> 78 - 60 = 18°

is cos108 + sin18 = 0 ???
YES

ok, then , let's prove it

LS = cosxcosπ/6 - sinxsinπ/6 + sinxcosπ/3 - cosxsinπ/3
= √3/2 cosx -1/2 sinx + 1/2 sinx - √3/2 cosx
= 0
= RS

Answer 2

or, since we know that cos(pi/2 - x) = sin(x)

cos(pi/2 - (x+pi/6)) = sin(pi/3 - x) = -sin(x-pi/3)