If x=a cos Φ, y=b sin Φ show that d^2y/dx^2=-b/a^2 csc^3Φ

Question

Under derivatives in calculus trigonometric questions in finding calculus

Answer 1

x/y=(a/b)cotø.

x'=-asinø, y'=bcosø, dy/dx=y'/x'=-(b/a)cotø=-(b^2/a^2)x/y.

But x^2=a^2cos^2(ø) and y^2=b^2sin^2(ø)=b^2(1-x^2/a^2).

d2y/dx2 = -(b/a)^2(y-xdy/dx)/y^2 = -(b/a)^2(bsinø-acosø(-b/a)cotø)/(bsinø)^2 =

-(b/a^2)(sin^2(ø)+cos^2(ø))/sin^3(ø)=-(b/a^2)cosec^3(ø) QED