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Prove that:tan^(-1)sqrt(x)=1/2cos^(-1)((...

Prove that:`tan^(-1)sqrt(x)=1/2cos^(-1)((1-x)/(1+x)), x in [0,1]`

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Let `sqrt(x)=tan thetaimpliestan^(2)theta`
`:. RHS=(1)/(2) cos^(-1)""((1-x)/(1+x)), " " x in [0,1]`
`=(1)/(2) cos^(-1)""((1-tan^(2) theta)/(1+tan^(2)theta))`
`=(1)/(2)cos^(-1)(cos2theta)=(1)/(2)(2theta)`
`=theta=tan^(-1)sqrt(x)=` LHS Hence Proved.
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