Question

Find $\frac{dy}{dx}$,
$x=\sin t,y=\cos 2t$

Answer 1

$x=\sin t$

$\frac{dx}{dt}=\frac{d}{dt}\left(\sin t\right)$

$=\cos t$

$y=\cos 2t$

$\frac{dy}{dt}=\frac{d}{dt}\left(\cos 2t\right)$

$=-\sin 2t.\frac{d}{dt}\left(2t\right)$

$=-\sin 2t.2$

$=-2\sin 2t$

$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}$

$=\frac{dy}{dt}\times \frac{1}{\frac{dx}{dt}}$

$=-2\sin 2t\times \frac{1}{\cos t}$

$=\frac{-2\sin 2t}{\cos t}$

$=\frac{-2\left(2\sin t\cos t\right)}{\cos t}$ $(\because \sin 2\theta =2\sin \theta \cos \theta )$

$\frac{dy}{dx}=-4\sin t$