Clarification on Implicit derivatives

so given the question:

sin(x+y)+sin(x-y)=1
I expanded and found 2sinxcosy =1
so my question is this:
does the derivative of (2sinxcosy=1) Equal the derivative of (cosy=1/(2sinx)) - all derivative of y with respect to x

I think they should be equal, because 2sinxcosy=1 is the same as cosy=1/(2sinx)
but when I do the derivative of each case, I get 2 different answers...

so what am i doing wrong... and if they don't equal, then which implicit derivative should i take???

Let's differentiate both
cosy=1/(2sinx)
-siny dy/dx = -1/(2sin^2x). cosx
dy/dx=cos x/(2sin^2x.siny)
But using cosy=1/2sinx, we get
dy/dx=cosxcosy/sinxsiny
and the other method:
2sinxcosy=1
2sinx(-siny dy/dx) + 2cosxcosy = 0
dy/dx=cosxcosy/sinxsiny
same answer




dy/dx=cosxcosy/sinxsiny