Prove or disprove?
posted by Jerry on .
cos(X+pi/6) + sin(xpi/3) = 0
I am working with one side but I am not sure if I can subtract at all here.

To see if it is an identity, pick any value of x
e.g. x = 78°
then x+π/6 > 78+30 = 108°
and xπ/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 
or, since we know that cos(pi/2  x) = sin(x)
cos(pi/2  (x+pi/6)) = sin(pi/3  x) = sin(xpi/3)