InterviewSolution
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InterviewSolution
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Differentiate `tan^(-1){(sqrt(1+x^2)-sqrt(1-x^2))/(sqrt(1+x^2)+sqrt(1-x^2))}`with respect to `cos^(-1)x^2` |
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Answer» Let `u=tan^(-1){(sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2)))} and v = cos^(-1)x^(2).` Then, `cos^(-1)x^(2)=v rArr x^(2)=cos v`. Putting `x^(2)=cos v`, we get `u=tan^(-1){(sqrt(1+cosv)-sqrt(1-cosv))/(sqrt(1+cosv)+sqrt(1-cosv))}` `rArrtan^(-1){(sqrt(2cos^(2)(v//2))-sqrt(2sin^(2)(v//2)))/(sqrt(2cos^(2)(v//2))+sqrt(2sin^(2)(v//2)))}` `=tan^(-1){(cos(v//2)-sin(v//2))/(cos(v//2)+sin(v//2))}=tan^(-1){(1-tan(v//2))/(1+tan(v//2))}` `" [dividing num. and denom. by cos (v/2)]"` `=tan^(-1){tan((pi)/(4)-(v)/(2))}=((pi)/(4)-(v)/(2)).` `thereforeu=((pi)/(4)-(v)/(2))rArr (du)/(dv)=(-1)/(2).` |
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