Find dy/dx in terms of t for the curve given by the parametric equations x = tan(t) , y = sec(t) for -pi/2<t<pi/2.

We know that dy/dx = (dy/dt) * (dt/dx). Differentiating each of the equations with respect to t gives. dy/dt = sec(t) tan(t) and dx/dt = sec²(t). Since dt/dx = 1 / (dx/dt) we have that dt/dx = 1/(sec²(t)) = cos²(t). Substituting back into the first equation gives dy/dx = cos²(t) sec(t) tan(t) . Using the following identities. sec(t) = 1/cos(t) and tan(t) = sin(t)/cos(t) .dy/dx = sin(t)*(cos²(t)/cos²(t))= sin(t). So the final answer is dy/dx = sin(t)