Verified answer

f(x,y) = cos x sin y ==> f(π/3, π/2) = 1/2.

f_x = -sin x sin y ==> f_x (π/3, π/2) = -√3/2

f_y = cos x cos y ==> f_y (π/3, π/2) = 0

f_xx = -cos x sin y ==> f_xx(π/3, π/2) = -1/2.

f_xy = -sin x cos y ==> f_xy(π/3, π/2) = 0

f_yy = -cos x sin y ==> f_yy (π/3, π/2) = -1/2

So, we have

T₂(x, y) = 1/2 + (-√3/2)(x - π/3) + 0(y - π/2) + (1/2!)[(-1/2)(x - π/3)^2 + 2 * 0(x - π/3)(y - π/2) + (-1/2)(y - π/2)^2]

= 1/2 - (√3/2)(x - π/3) - (1/4)(x - π/3)^2 - (1/4)(y - π/2)^2.

I hope this helps!