$x = a\left( {\cos t + \log \tan \cfrac{t}{2}} \right),y = a\sin t$

Here, $x = a\left( {\cos t + \log \tan \cfrac{t}{2}} \right)$ …(1)
and $y = a\sin t$ …(2)
Differentiating (1) \& (2) w.r.t. t, we get

$\cfrac{{dx}}{{dt}} = a\left[ { - \sin t + \cfrac{1}{{\tan \cfrac{t}{2}}}\cfrac{d}{{dt}}\left( {\tan \cfrac{t}{2}} \right)} \right]$

$= a\left[ { - \sin t + \cfrac{1}{{\tan \cfrac{t}{2}}}{{\sec }^2}\cfrac{t}{2} \cdot \cfrac{1}{2}} \right] = \cfrac{{a{{\cos }^2}t}}{{\sin t}}$
$\cfrac{{dy}}{{dt}} = a\cos t$

therefore, $\cfrac{{dy}}{{dx}} = \cfrac{{dy/dt}}{{dx/dt}} = \cfrac{{a\cos t\sin t}}{{a{{\cos }^2}t}} = \tan t$