Question

If $x$ and $y$ are connected parametrically by the equation,without eliminating the parameter,find $\frac{dy}{dx}$.
$x=a(cos\,t+log\,tan\frac{t}{2}),y=a\,sint$

Answer 1

The correct answer is $tan\,t$
The given equations are $x=a(cos\,t+log\,tan\frac{t}{2}),y=a\,sint$
Then,$\frac{dx}{dt}=a[\frac{d}{dt}(cost)+\frac{d}{dt}(log\,tan\frac{t}{2})]$
$=a[-sin\,t+\frac{1}{tan\frac{t}{2}}.\frac{d}{dt}(tan\frac{t}{2})]$
$=a[-sin\,t+cot\frac{t}{2}.sec^2\frac{t}{2}.\frac{d}{dt}(\frac{t}{2})]$
$=a[-sint+\frac{cos\frac{t}{2}}{sin\frac{t}{2}}\times \frac{1}{cos^2\frac{t}{2}}.\frac{1}{2}]$
$=a[-sint+\frac{1}{2sin\frac{t}{2}cos\frac{t}{2}}]$
$=a(-sint+\frac{1}{sint})$
$=a(\frac{-sin^2t+1}{sint})$
$=a\frac{cos^2t}{sint}$
$\frac{dy}{dt}=a\frac{d}{dt}(sint)=acost$
$∴\frac{dy}{dx}=\frac{(\frac{dy}{dt})}{(\frac{dx}{dt})}=\frac{acost}{(\frac{acos^2t}{sint})}$
$=\frac{sint}{cost}=tan\,t$