If $x = 2 cos t - c o s 2 t, y = 2 sin t - sin 2 t,$ find the value of $d y / d x$ .

tan \frac{3 t}{2}

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tan \frac{- 3 t}{2}

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tan \frac{3 t}{4}

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tan \frac{- 3 t}{4}

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Solution

The correct option is A $tan \frac{3 t}{2}$
$\frac{d x}{d t} = \frac{d}{d t} (2 cos (t) - cos (2 t))$
$= \frac{d}{d t} (2 cos (t)) - \frac{d}{d t} (cos (2 t))$
$= - 2 sin (t) - (- 2 sin (2 t))$
$= - 2 sin (t) + 2 sin (2 t)$

$\frac{d y}{d t} = \frac{d}{d t} (2 sin (t) - sin (2 t))$
$= \frac{d}{d t} (2 sin (t)) - \frac{d}{d t} (sin (2 t))$
$= 2 cos (t) - cos (2 t) \cdot 2$

$\frac{d y}{d x} = \frac{d y}{d t} \times \frac{d t}{d x} = \frac{c o s (t) - c o s (2 t)}{- s i n (t) + s i n (2 t)} = \frac{2 s i n (\frac{3 t}{2}) s i n t}{2 c o s (\frac{3 t}{2}) s i n (t)} = t a n (\frac{3 t}{2})$

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