Question

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

Answer 1

Given,

$x=\sin t,y=\cos 2t$ $......\left(1\right)$

Differentiating this equation with respect to ‘t’ and we get,

$\frac{dx}{dt}=\frac{d}{dt}\sin t$, $\frac{dy}{dt}=\frac{d}{dt}\cos 2t$

$\frac{dx}{dt}=\cos t$ ,

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

$\frac{dx}{dt}=\cos t$ ,

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

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

Put the value of equation (2) in $\frac{dy}{dx}$, and we get,

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

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

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

$=-2\times 2\sin t$

$=-4\sin t$

Hence, It is required solution.