Please help me to find an equation for the tangent to the curve xsin2y=ycos2x at the point (phi/4, phi/2)

Answer 1

first, it's PI, not PHI
pi(π) = 3.14, the ratio of circumference to diameter
phi(φ) = 1.62 (1+√5)/2, the golden ratio

So, we use implicit differentiation to get

xsin2y = ycos2x
sin2y + 2xcos2y y' = cos2x y' - 2ysin2x

y' = -(sin2y + 2y sin2x)/(2x cos2y - cos2x)

at (π/4,π/2), that gives a slope of

y' = -(0 + π*(1))/(π/2*(-1) - 0)
= -π/(-π/2)
= 2

So, bow you want the line with slope=2 through (π/4,π/2):

y - π/2 = 2(x - π/4)
y = 2x

Answer 2

xsin2y=ycos2x