सिद्ध कीजिए कि
` tan ^( -1) sqrt x = (1)/(2) cos ^(-1) ((1 -x )/(1+x )) `

Explanation

R.H.S. ` = (1)/(2) cos ^(-1) (( 1 -x ) /( 1 + x )) `
माना ` sqrt x = tan theta `
` rArr x = tan ^( 2) theta `
` rArr (1 - x ) /( 1 + x ) = ( 1 - tan ^ 2 theta ) /( 1 + tan ^ 2 theta ) = cos 2 theta `
` therefore ` R.H.S. = ` ( 1 ) /(2) cos ^(-1) ( cos 2 theta ) = (1 ) /(2) xx 2 theta = theta `
` = tan ^( -1) (sqrt x ) , " " [ because sqrt x = tan theta rArr tan ^( -1) (sqrt x ) = theta ] `
` = tan ^( -1) (sqrt x ) `
= L.H.S.