Question

If \( x = a\left\{ \cos t + \frac{1}{2}\log\!\left(\tan^2\frac{t}{2}\right) \right\} \) and \( y = a\sin t \), then \( \frac{dy}{dx} = \) __________.

• A. \( \frac{a^2\cos^3 t}{\sin t} \)
• B. \( \tan t \)
• C. \( a\tan t \sec t \)
• D. \( \sec^2 t \)

Hint:

For parametric equations, always use \( \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \)Differentiate both variables with respect to the parameter first.

Answer 1

The Correct Option is B

Step 1: Write the parametric equations.
\[ x = a\left\{\cos t + \frac{1}{2}\log\left(\tan^2\frac{t}{2}\right)\right\} \]
\[ y = a\sin t \]

Step 2: Differentiate \( y \) with respect to \( t \).
\[ \frac{dy}{dt} = a\cos t \]

Step 3: Simplify logarithmic term.
\[ \frac{1}{2}\log\left(\tan^2\frac{t}{2}\right) = \log\left(\tan\frac{t}{2}\right) \]

Step 4: Differentiate \( x \) with respect to \( t \).
\[ \frac{dx}{dt} = a\left[-\sin t + \frac{d}{dt}\log\left(\tan\frac{t}{2}\right)\right] \]
Now,
\[ \frac{d}{dt}\log\left(\tan\frac{t}{2}\right)=\frac{1}{\sin t} \]
So:
\[ \frac{dx}{dt}=a\left(-\sin t+\frac{1}{\sin t}\right) \]

Step 5: Simplify \( \frac{dx}{dt} \).
\[ \frac{dx}{dt} = a\left(\frac{1-\sin^2 t}{\sin t}\right) \]
\[ = a\left(\frac{\cos^2 t}{\sin t}\right) \]

Step 6: Use parametric derivative formula.
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]
\[ = \frac{a\cos t}{a\frac{\cos^2 t}{\sin t}} \]

Step 7: Final conclusion.
\[ \frac{dy}{dx} = \frac{\sin t}{\cos t} = \tan t \]
\[ \boxed{\tan t} \]