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InterviewSolution
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सिद्ध कीजिए 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» हमें सिद्ध करना है 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)` माना `(1)/(2) cos^(-1)x^(2)=thetaimplies x^(2)=cos2 theta` `:. sqrt(1+x^(2))=sqrt(1+cos2 theta)` `=sqrt(2cos^(2)theta)=sqrt(2)cos theta` तथा ` sqrt(10-x^(2))=sqrt(1-cos2 theta)=sqrt(2 sin^(2)theta)=sqrt(2)sin theta ` अब , `(sqrt(1+x^(2))+sqrt(1+x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))=(sqrt(2)cos theta+sqrt(2)sin theta )/(sqrt(2) cos theta -sqrt(2)sin theta)` `=( cos theta + sin theta )/( cos theta - sin theta )=(1+ tan theta )/(1- tan theta )=(tan""(pi)/(4)+tan theta )/(1-tan""(pi)/(4)tan theta )` `(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))=tan((pi)/(4)+theta)` `implies tan^(-1)[(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))]=tan""(pi)/(4)+theta` `implies tan^(-1)[(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))]=tan""(pi)/(4)+(1)/(2)cos^(-1)x^(2)` |
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