X=t- tan t y=ln(cos t) show that

Question

dy/dx=cot t?

Answer 1

The parametric equation : x = t - tan t and y = ln(cos t).

dy/dt = (1/cos t)(- sin t)

= - sin t/cos t

= - tan t.

dx/dt = 1 - sec² t

= 1 - (1 + tan² t)

= 1 - 1 - tan² t

= - tan² t.

• dy/dx = (dy/dt)/(dx/dt)

Plug the values dy/dt = - tan t and dx/dt = - tan² t.

= - tan t / - tan² t

= 1/tan t

= cot t.

Therefore, dy/dx = cot t.               (hence proved)