Calculus: Early Transcendentals, Chapter 3, 3.5, Section 3.5, Problem 26

Note:- 1) If y = sinx ; then dy/dx = cosx ;
2) If y = u + v ; where both u & v are functions of 'x' , then
dy/dx = (du/dx) + (dv/dx)
3) If y = k ; where 'k' = constant ; then dy/dx = 0
4) cos(2pi) = 1
Now, the given function is :-
sin(x+y) = 2x - 2y
or, cos(x+y)*[1 + (dy/dx)] = 2 - 2*(dy/dx)
Thus, putting x = y = pi ; we get
cos(2pi)*[1 + (dy/dx)] = 2 - 2*(dy/dx)
or, [1 + (dy/dx)] = 2 - 2*(dy/dx)
or, dy/dx = 1/3 = slope of the tangent to the curve at (pi,pi)
Thus, equation of the tangent line to the given curve at the point (pi,pi) is :-
y - pi = (1/3)*(x - pi)
or, 3y - 3pi = x - pi
or, 3y = x + 2pi = equation of the tangent