InterviewSolution
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InterviewSolution
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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)` |
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Answer» `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)]` |
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