सिद्ध करें कि
`tan^(-1)[(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))] = (pi)/(4)+(1)/(2) cos^(-1) x^(2)`

Explanation

`L.H.S. = tan^(-1)((1+sqrt((1-x^(2))/(1+x^(2))))/(1-sqrt((1-x^(2))/(1+x^(2)))))" "...(1)`
`x^(2) = cos 2theta`, रखने पर, `sqrt((1-x^(2))/(1+x^(2)))=sqrt((1-cos2theta)/(1+cos 2theta))`
` = sqrt((2 sin^(2)theta)/(2 cos^(2)theta))=|tan theta|= tantheta[because 0 lesqrt((1-x^(2))/(1+x^(2)))le1therefore 0 letheta le(pi)/(4)]`
अब,(1) से. `L.H.S. = tan^(-1)((1+tantheta)/(1-tantheta))`
` = tan^(-1)[tan((pi)/(4)+theta)] =(pi)/(4)+theta=(1)/(2) cos^(-1) x^(2)`
`[because cos 2theta = x^(2) therefore2 theta = cos^(-1) x^(2)therefore theta = (1)/(2) cos^(-1) x^(2)]`