Question

Find `(dy)/(dx)` if x = a cosec θ, y = b cot θ at θ = `π/4`

Answer 1

x = a cosec θ, y = b cot θ
Consider, y = b cot θ
Differentiating w.r.t. θ

`(dy)/(dθ) = b(-cosec^2θ)`

       = `-b cosec^2θ`

Consider, x = a cosec θ
Differentiating w.r.to θ

`(dy)/(dθ)` = a(-cosec θ. cot θ)

       = -a cosec  θ. cot θ

Now `(dy)/(dx)` = `((dy)/(dθ))/((dy)/(dθ))``

                        = `(-b cosec^2θ)/(-a cosec θ . cot θ)`

                        = `b/a  (cosecθ)/(cotθ)`

`(dy/dx)_(at θ = π/4) = b/a (cosec (π/4))/cot (π/4)`

                       = `b/a (sqrt2/1)`

                       = `sqrt2 b/a`