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