[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]
[tex]y=\cos t\implies\dfrac{\mathrm dy}{\mathrm dt}=-\sin t[/tex]
[tex]x=\cos2t\implies\dfrac{\mathrm dx}{\mathrm dt}=-2\sin2t=-4\sin t\cos t[/tex]
[tex]\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{-\sin t}{-4\sin t\cos t}=\dfrac14\sec t[/tex]
Let [tex]z=\dfrac{\mathrm dy}{\mathrm dx}[/tex]. Then
[tex]\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm dz}{\mathrm dx}=\dfrac{\mathrm dz}{\mathrm dt}\dfrac{\mathrm dt}{\mathrm dx}=\dfrac{\frac{\mathrm dz}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]
[tex]\implies\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\frac{\mathrm d}{\mathrm dt}\left[\frac14\sec t\right]}{-4\sin t\cos t}=\dfrac{\frac14\sec t\tan t}{-4\sin t\cos t}=-\dfrac1{16}\sec^3t[/tex]
