# If $x$ and $y$ are connected parametrically by the equations given in $x = \sin t, y = \cos 2t$ without eliminating the parameter, Find $\large\frac{dy}{dx}$

$\begin{array}{1 1} -4\cos t \\ -5\sin t \\ 4\sin t \\ -4\sin t \end{array}$

Toolbox:
• By chain rule we have $\large\frac{dy}{dx}=\frac{dy}{dt}$$\times\large\frac{dt}{dx} Step 1: Given : x=\sin t Differentiating with respect to t \large\frac{dx}{dt}$$=\cos t$
$f'(t)=\cos t$
Step 2:
$y=\cos 2t$
Differentiating with respect to $t$
$\large\frac{dy}{dt}$$=-\sin 2t.2 \large\frac{dy}{dt}$$=-2\sin 2t$
$g'(t)=-2\sin 2t$
Step 3:
$\large\frac{dy}{dx}=\Large\frac{\Large\frac{dy}{dt}}{\Large\frac{dx}{dt}}$
$\large\frac{dy}{dx}=\frac{dy}{dt}$$\times\large\frac{dt}{dx} \quad\;=-2\sin 2t\times \large\frac{1}{\cos t} We know that \sin 2A=2\sin A\cos A Hence \sin 2t=2\sin t\cos t Replacing \sin 2t we get \large\frac{dy}{dt}$$=-4\sin t\cos t$