Question

If y = cos^–1 x, Find ${\displaystyle \frac{d^{2}y}{{dx}^2}}$ in terms of y alone.

Answer 1

Given, y = cos^-1 x

⇒ x = cos y

Differentiating both sides with respect to x,

${\displaystyle \frac{d}{dx}\left(x\right)=\frac{d}{dx}\cos y}$

or ${\displaystyle 1=-\sin y \frac{dy}{dx}}$  

${\displaystyle \Rightarrow \frac{dy}{dx}=-\frac{1}{\sin y}=-\cos ec y}$

Differentiating both sides again with respect to x,

${\displaystyle \frac{d^{2}y}{{dx}^2}=-\frac{d}{dx}}$  cosec  y = - (- cosec  y cot y) ${\displaystyle \frac{dy}{dx}}$

= cosec y cot y (- cosec y)       ...${\displaystyle [\frac{dy}{dx}}$ substituting the value of]

= - cosec² y cot y