Given-implicit-expression-x-tan-y-tan-where-x-y-and-is-variable-Prove-that-dy-dx-1-y-2-1-x-2-0-

Question Number 121633 by liberty last updated on 10/Nov/20

Given implicit expression {\begin{cases} x = \tan α \\ y = \tan β \end{cases}

where x, y, α and β is variable . Prove that

\frac{dy}{dx} + \frac{1 + y^{2}}{1 + x^{2}} = 0 .

Commented by Dwaipayan Shikari last updated on 10/Nov/20

x = t a n α \Rightarrow {t a n}^{- 1} x = α \Rightarrow \frac{1}{1 + x^{2}} = \frac{d α}{d x}

y = t a n β \Rightarrow {t a n}^{- 1} y = β \Rightarrow \frac{1}{1 + y^{2}} = \frac{d β}{d y}

\frac{d y}{d x} . \frac{d α}{d β} = \frac{1 + y^{2}}{1 + x^{2}}

Commented by liberty last updated on 10/Nov/20

not proved ?

Commented by Dwaipayan Shikari last updated on 10/Nov/20

W e h a v e t o k n o w r e l a t i o n b e t w e e n α a n d β

Commented by Dwaipayan Shikari last updated on 10/Nov/20

I f α + β = 0 t h e n

- \frac{d y}{d x} = \frac{1 + y^{2}}{1 + x^{2}} \Rightarrow \frac{d y}{d x} + \frac{1 + y^{2}}{1 + x^{2}} = 0