Question

If y = cos^–1 x, find `(d^2y)/dx^2` in terms of y alone.

Answer 1

Given, y = cos^–1 x

⇒ x = cos y

Differentiating both sides with respect to x,

`d/dx (x) = d/dx cos y`

`1 = - sin y dy/dx`  

`dy/dx = - 1/sin y`

`dy/dx` = −cosec y

Differentiating both sides again with respect to x,

`(d^2 y)/dx^2 = - d/dx` cosec y

= −(−cosec y cot y) `dy/dx`

= [cosec y cot y] (−cosec y) ...`["Substituting the value of" dy/dx]`

= −cosec² y cot y