If `x=a(theta+sintheta),y=a(1+costheta),y=a(1+costheta),` Prove that `(d^2y)/(dx^2)=-a/(y^2)`

Explanation

`x=a(theta+sin theta), y=a(1+cos theta)`


to prove `frac{d^{2} y}{d x^{2}}=-frac{a}{y^{2}}`


`frac{d x}{d theta}=a(1+cos theta)`


`frac{d y}{d theta}=a(-sin theta)`


`frac{d y}{d x}=-frac{sin theta}{1+cos theta}=-frac{2 sin theta cos theta}{2 cos ^{2} frac{theta}{2}}`


`=-tan frac{theta}{2}`


`frac{d^{2} y}{d x^{2}}=-sec ^{2} theta cdot frac{1}{2} cdot frac{d theta}{d x}`


`( y=a(1+cos theta)=a times 2 cos ^{2} frac{theta}{2}, y=frac{2 a}{sec ^{2} theta}, therefore sec ^{2} theta=frac{2 a}{y})`


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