[m]x`_{t}=((1+cos^2t)^2)`_{t}=2\cdot (1+cos^2t)\cdot (1+cos^2t)`_{t}==2\cdot (1+cos^2t)\cdot 2cost\cdot (cost)`_{t}=[/m]
[m]=-4 sint cost\cdot (1+cos^2t)=-2 sin2t \cdot (1+cos^2t)[/m]
[m]y`_{t}=(cost)`_{t}=-sint[/m]
[m]y``_{xx}=\frac{y``_{tt}\cdot x`_{t}-y`_{t}\cdot x``_{tt}}{x`_{t}}[/m]
[m]x``_{tt}=(x`_{t})`_{t}=(-2 sin2t \cdot (1+cos^2t))`_{t}=(-2sin2t)`_{t}\cdot (1+cos^2t)+(-2sint)\cdot (1+cos^2t)`_{t}=[/m]
[m]=-2\cdot (cos2t)\cdot 2\cdot (1+cos^2t)-2sint\cdot 2cost\cdot (-sint)=-8\cdot cos2t\cdot (1+cos^2t)+4sin^2t\cdot cost[/m]
[m]y``_{tt}=(y`_{t})`_{t}=(-sint)`_{t}=-cost[/m]
[m]y``_{xx}=\frac{-cost\cdot (-2 sin2t \cdot (1+cos^2t))-(-sint)\cdot(-8\cdot cos2t\cdot (1+cos^2t)+4sin^2t\cdot cost)}{(-2 sin2t \cdot (1+cos^2t))^3}[/m]
О т в е т.
[m]y``_{xx}=\frac{2 sin2t \cdot cost\cdot (1+cos^2t)-8\cdot sint\cdot cos2t \cdot (1+cos^2t)-4sin^3t\cdot cost}{(-8 )\cdot sin^32t \cdot (1+cos^2t)^3}[/m]