InterviewSolution
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InterviewSolution
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Prove that`tan^(-1)((sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x)))=pi/4-1/2cos^(-1)x,-1/(sqrt(2))lt=xlt=1` |
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Answer» `L.H.S. = tan^-1[(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]` `=tan^-1[(sqrt(1+x)(1-sqrt(1-x)/sqrt(1+x)))/(sqrt(1+x)(1+sqrt(1-x)/sqrt(1+x)))]` `=tan^-1[(1-sqrt(1-x)/sqrt(1+x))/(1+sqrt(1-x)/sqrt(1+x))]` As, `tan^-1x-tan^-1y = (x-y)/(1+xy)` So, our expression becomes, `=tan^-1(1)+tan^-1(sqrt(1-x)/sqrt(1+x))` `=pi/4+1/2(2tan^-1(sqrt(1-x)/sqrt(1+x)))` Also, `2tan^-1y = cos^-1((1-y^2)/(1+y^2))` So, our expression becomes, `= pi/4+1/2cos^-1((1-(sqrt(1-x)/sqrt(1+x))^2)/(1+(sqrt(1-x)/sqrt(1+x))^2))` `=pi/4+1/2cos^-1((1+x-1+x)/(1+x+1-x))` `=pi/4+1/2cos^-1((2x)/2)` `=pi/4+1/2cos^-1(x)=R.H.S.` |
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