Question

If x and y are connected parametrically by the equation, without eliminating the parameter, find ${\displaystyle \frac{dy}{dx}.}$

${\displaystyle x=a(\cos t+\log \tan \frac{t}{2}),y= a\sin t}$

Answer 1

x = a ${\displaystyle (\cos t+\log \tan \frac{t}{2})y=a\sin t}$

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

${\displaystyle =a[-\sin t+\frac{1}{\sin t}]\&\frac{dy}{dt}=a\cos t}$

${\displaystyle =\frac{dy}{dt}=\frac{dy}{dt}÷\frac{dx}{dt}}$

${\displaystyle =\frac{a\cos t}{a\frac{{\cos }^{2}t}{\sin t}}}$

= tan t