Question

Find : ` d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`

Answer 1

`y= d/dx cos^−1 ((x−x^(−1))/(x+x^(−1)))`

`= d/dx cos^−1 ((x^2−1)/(x^2+1))`

`Let x=tanθ`

`∴y=cos^(−1) ((tan^2θ−1)/(tan^2θ+1))`

`=cos^(-1)(-(1-tan^2theta)/(1+tan^2theta))`

`=pi - cos^(-1) (cos 2theta) [cos^(-1)(-x)=pi- cos^(-1)x]`

`=π−2θ`

`=π−2tan^(−1)x`

Differentiating both sides w.r.t. x, we have

`dy/dx=0-2xx1/(1+x^2)`

`therefore d/dx cos^-1 ((x-x^(-1))/(x+x^(-1)))=-2/(1+x^2)`