Hyperbola and inclination

Let P=(x0,y0), then tangent at P is
L: (x0/a^2) x - (y0/b^2) y = 1
Slope = m0 = b^2x0/a^2y0
Denote S = (-c,0) and Q = (c,0) , then
Slope of SP = m1 = y0/(x0+c) and Slope of QP = m2 = y0/(x0-c)

By using tan A = (m1-m2)/(1+m1m2), where A denote the angle between the 2 lines.
You can check that:
(m1-m0)/(1+m1m0) = (m0-m2)/(1+m0m2) = b^2/cy0
[You need the substitution c^2 = a^2+b^2 which is default in hyperbola.]
As a result, the tangent of the angles between SP and L, L and QP are same.
=> Q.E.D.