InterviewSolution
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InterviewSolution
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सिद्ध कीजिए कि ` cos[tan^(-1){sin(cot^(-1)x)}]=sqrt((x^(2)+1)/(x^(2)+2))` |
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Answer» सर्वप्रथम माना `cot^(-1)x=theta` `implies x= cot theta ` `:. " cosec" theta=sqrt(1+cot^(2)theta)=sqrt(1+x^(2))` ` implies sin theta = (1)/(sqrt(1+x^(2)))` `implies sin (cot^(-1)x)=(1)/(sqrt(1+x^(2)))` `= tan^(-1)[sin(cot^(-1)x)]=tan^(-1)""(1)/(sqrt(1+x^(2)))= phi` (say) तब `, cos[tan^(-1){sin(cot^(-1)x)}]=cos phi" "`......(1) अब , चूँकि `tan^(-1)""(1)/(sqrt(1+x^(2)))=phi` , तब ` tan phi=(1)/(sqrt(1+x^(2)))` `:. sec phi=sqrt(1+tan^(2)phi)=sqrt(1+(1)/((1+x^(2))))=sqrt((x^(2)+1)/(x^(2)+2))` `:. cos phi= sqrt((1+x^(2))/(2+x^(2)))" " `.....(2) अतः समीकरण ( 1 ) व ( 2 ) से , ` cos[tan^(-1){sin(cot^(-1)x)}]=sqrt(x^(2)+1)/(x^(2)+2)` |
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