first I've transformed your equation into:
x(t) = t*cos(t)
y(t) = -t*sin(t)
Then the point T where the tangent touches the curve has the coordinates:
To calculate the tangent you need the gradient of the curve in T. I use the formula:
dy/dx = (dy/dt) / (dx/dt). Use product rule:
dy/dx = (sin(t)*(-1) + (-t)*cos(t)) / (cos(t) + t*(-sin(t)))
Now plug in t = π/2
therefore the slope of the tangent is 2/π.
Use the point-slope-formula to get the equation of the straight line:
t: y = (2/π)*x-(π/2)
I've attached an image of your curve and the tangent.