Question

If x=a(t-sin t), y=a(1-cos t),show that dy /dx=cot t/2

Answer 1

Given that,
x=a(t-sin t)  .........(1)
y=a(1-cos t)  ............. (2)
We have to prove that dy / dx=cot t/2
Solution:
Differentiate (1) with respect to 't' , we get:
x=a(t-sin t) 
dx/dt = a (1 - cos t)    (∵ d/dt (sin t) = cos t )
Now differentiate (2) with respect to 't' , we get:
y=a(1-cos t)
dy/dt = - ( - sin t )    (∵ d/dt (cos t) = - sin t )
dy/dt = a . sin t
Now,
dy/dx = (dy/dt) . (dt/dx)
Substituting values, we get:
dy/dx = a (sin t) . 1 / a (1-cos t)
dy/dx = a (sint) / a (1 - cos t)
dy/dx = sin t / 1 - cos t
dy/dx = (2 . sin t/2 cos t/2) / 2 sin² t/2
dy/dx = (sin t/2 cos t/2) / sin² t/2
dy/dx = cos t/2 / sin t/2
dy/dx = cot t/2   (∵ cosФ/sinФ = cotФ )
Hence proved.
Hopefully it helps. Thanks.