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Prove thattan^(-1)((sqrt(1+x)-sqrt(1-x))...

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`

Text Solution

Verified by Experts

`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))`
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