InterviewSolution
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InterviewSolution
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यदि `cos^(-1)x+cos^(-1)y+cos^(-1)z=pi`, सिद्ध कीजिए कि `x^(2)+y^(2)+z^(2)+2xyz=1` |
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Answer» माना `cos^(-1)x=A, cos^(-1)y=B , cos^(-1)z=C` `implies cos A =x , cos B=y,cos C=z` `:. A+B+C=piimplies A+B=pi-C` `implies cos(A+B)=cos(pi-C)` `implies cosA cos B- sinA sin B=-cosC` `impliescosA cosB-sinA sinB+cosC=0` `implies cosA cosB + cosC=sinA sin B` `implies cosA cosB +cosC=sqrt(1-cos^(2)A)sqrt(1-cos^(2)B)` `implies xy+z=sqrt(1-x^(2))sqrt(1-y^(2))` `implies (xy)^(2)+z^(2)+2xyz=(1-x^(2))(1-y^(2))` ( दोनों ओर का वर्ग करने पर ) `implies x^(2)y^(2)+z^(2)+2xyz=1-y^(2)-x^(2)+x^(2)y^(2)` `implies x^(2)+y^(2)+z^(2) +2xyz=1` |
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