Calculus: Early Transcendentals, Chapter 3, 3.5, Section 3.5, Problem 25

Note:- 1) If y = sinx ; then dy/dx = cosx
2) If y = cosx ; then dy/dx = -sinx
3) If y = u*v ; where both u & v are functions of 'x' , then
dy/dx = u*(dv/dx) + v*(du/dx)
4) If y = k ; where 'k' = constant ; then dy/dx = 0
5) cos(pi/2) = sin(pi) = 0
6) sin(pi/2) = 1 & cos(pi) = -1
Now, the given function is :-
y*sin(2x) = x*cos(2y)
Differentiating both sides w.r.t 'x' we get
{sin(2x)}*(dy/dx) + 2y*cos(2x) = cos(2y) - 2x*{sin(2y)}*(dy/dx)
Putting x = pi/2 & y = pi/4 in the above equation we get
dy/dx = slope of the tangent at point (pi/2,pi/4) = 1/2
Thus, the eqaution of the tangent at the point (pi/2,pi/4) is:-
y - (pi/4) = (1/2)*{x - (pi/2)}
y = (1/2)*x = tangent equation