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