Question

$x=\frac{{\sin }^{3}t}{\sqrt{\cos 2t}},y=\frac{{\cos }^{3}t}{\sqrt{\cos 2t}}$

Answer 1

Let,

$x=\frac{{\sin }^{3}t}{\sqrt{\cos 2t}}$ …..(1)

and

$y=\frac{{\cos }^{3}t}{\sqrt{\cos 2t}}$ …(2)

Differentiation with respect to $t$ of equation (1) we get

$\frac{dx}{dt}=\frac{d}{dx}\frac{{\sin }^{3}t}{\sqrt{\cos 2t}}$

By quotient rule

$\frac{dx}{dt}=\frac{\sqrt{\cos 2t}.\frac{d}{dt}si{n}^{3}t-{\sin }^{3}t\frac{d}{dt}\sqrt{\cos 2t}}{{\left(\sqrt{\cos 2t}\right)}^2}$

$=\frac{\sqrt{\cos 2t}.\frac{d}{dt}3si{n}^{2}t\frac{d}{dt}\sin t-{\sin }^{3}t\frac{1}{2}{\left(\cos 2t\right)}^{\frac{-1}{2}}\frac{d}{dt}\cos 2t}{\cos 2t}$

$=\frac{\sqrt{\cos 2t}\left(3si{n}^{2}t\cos t\right)-\frac{{\sin }^{3}t}{2\sqrt{\cos 2t}}(-2\sin 2t)}{\cos 2t}$

$\frac{dx}{dt}=\frac{3{\sin }^{2}t\cos t\cos 2t+{\sin }^{3}t\sin 2t}{{\left(\cos 2t\right)}^{\frac{3}{2}}}$

$\frac{dx}{dt}=\frac{{\sin }^{2}t\cos t(3\cos 2t+2{\sin }^{2}t)}{{\left(\cos 2t\right)}^{\frac{3}{2}}}$

Differentiate equation $\left(2\right)$ with respect to $t$

$\frac{dy}{dt}=\frac{\sqrt{\cos 2t}.\frac{d}{dt}{\cos }^{3}t-{\cos }^{3}t\frac{d}{dt}\sqrt{\cos 2t}}{{\left(\sqrt{\cos 2t}\right)}^2}$

$\frac{dy}{dt}=\frac{\sqrt{\cos 2t}.3{\cos }^{2}t(-\sin t)-{\cos }^{3}t\left(\frac{1}{2\sqrt{\cos 2t}}(-2\sin 2t)\right)}{\cos 2t}$

$\frac{dy}{dt}=\frac{\cos 2t.3{\cos }^{2}t(-\sin t)+{\cos }^{3}t\left(2\sin t\cos t\right)}{{\cos }^{\frac{3}{2}}2t}$

$\frac{dy}{dt}=\frac{-3\cos 2t{\cos }^{2}t\left(\sin t\right)+{2\cos }^{4}t\left(\sin t\right)}{{\cos }^{\frac{3}{2}}2t}$

Therefore,

$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=$ $\frac{dy}{dx}=$ $\frac{\frac{-3\cos 2t{\cos }^{2}t\left(\sin t\right)+{2\cos }^{4}t\left(\sin t\right)}{{\cos }^{\frac{3}{2}}2t}}{\frac{{\sin }^{2}t\cos t(3\cos 2t+2{\sin }^{2}t)}{{\cos }^{\frac{3}{2}}2t}}$

$=\frac{\cos t(2{\cos }^{2}t-3(2{\cos }^{2}t-1))}{\sin t(3(1-2{\sin }^{2}t)+2{\sin }^{2}t)}$

$=\frac{\cos t(3-4{\cos }^{2}t)}{\sin t(3-4{\sin }^{2}t)}$

$=\frac{-(4{\cos }^{3}t-3\cos t)}{3\cos t-4{\cos }^{3}t}$

$=-\frac{\cos 3t}{\sin 3t}$

$=-\cot 3t$

Hence, the value is $-\cot 3t$.