If `x=asectheta,y=btantheta,"p r o v et h a t"(d^2y)/(dx^2)=-(b^4)/(a^2y^3)`

Explanation

`x=a sec theta y=b tan theta`


`frac{d x}{d theta}=a sec theta tan theta`


`frac{d y}{d theta}=b sec ^{2} theta`


`frac{(d y) / (d theta)}{(d x) / (d theta)}=frac{sec ^{2} theta}{sec theta tan theta} frac{b}{a}`


`frac{d y}{d x}=frac{sec theta b}{tan theta} frac{a}{a}`


`frac{d y}{d x}=frac{b}{a} {cosec} theta`


Differentiating w.r.t. `x`


`frac{d^{2} y}{d x^{2}}=frac{d({cosec} theta)}{d theta} times frac{d theta}{d x}`


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