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If $x$ and $y$ are connected parametrically by the given equations, without eliminating the parameter. Find $\frac{d y}{d x}$
$x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t$
$x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t$
Solution:
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Verified Answer
$\frac{d x}{d t}=a\left[-\sin t+\frac{1}{\tan \frac{t}{2}} \frac{d}{d t} \tan \frac{t}{2}\right]$
$=a\left[-\sin t+\frac{1}{\sin t}\right] \& \frac{d y}{d t}=a \cos t$
$\frac{d y}{d x}=\frac{d y}{d t} \div \frac{d x}{d t}=\frac{a \cos t}{a \frac{\cos ^2 t}{\sin t}}=\tan t$
$=a\left[-\sin t+\frac{1}{\sin t}\right] \& \frac{d y}{d t}=a \cos t$
$\frac{d y}{d x}=\frac{d y}{d t} \div \frac{d x}{d t}=\frac{a \cos t}{a \frac{\cos ^2 t}{\sin t}}=\tan t$
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