The coordinates of the point on the curve cos(y)=x-y, -pi<=x<=pi, where the tangent line to the curve is a vertical line is ____.
cos y = x - y
-sin y dy/dx = 1 - dy/dx
dy/dx - siny dy/dx = 1
dy/dx(1 - siny) = 1
dy/dx = 1/(1 - siny)
to have a vertical tangent, the slope of that tangent must be undefined, that is,
1 - siny = 0
sin y = 1
y = π/2 or y = -3π/2 or 5π/2
if y = π/2 , then
cos π/2 = x - π/2
0 = x-π/2
x = π/2 , so at point (π/2, π/2)
if y = -3π/2
cos(-3π/2) = x + 3π/2
x = -3π/2 , but that is outside our domain
if y = 5π/2
0 = x - 5π/2
x = 5π/2 , again, outside our domain
the only point within your domain is (π/2 , π/2)