Find $\cfrac{{dy}}{{dx}},ify = 12(1 - \cos t),x = 10(t - \sin t), - \cfrac{\pi }{2} < t < \cfrac{\pi }{2}.$

Here, $y = 12(1 - \cos t)$ ….(i)
$x = 10(t - \sin t)$ …(ii)

Differentiating (1) \ (2) w.r.t. t, we get
$\cfrac{{dy}}{{dt}} = 12[ - ( - \sin t)] = 12\sin t$

and $\cfrac{{dx}}{{dt}} = 10(1 - \cos t)$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = \cfrac{{dy/dt}}{{dx/dt}} = \cfrac{{12\sin t}}{{10(1 - \cos t)}} = \cfrac{{6\sin t}}{{5(1 - \cos t)}}$

$= \cfrac{6}{5}\left[ {\cfrac{{2\sin (t/2)\cos (t/2)}}{{2{{\sin }^2}(t/2)}}} \right] = \cfrac{6}{5}\cot (t/2)$