InterviewSolution
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InterviewSolution
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Prove that `cos (tan^(-1) (sin (cot^(-1) x))) = sqrt((x^(2) + 1)/(x^(2) + 2))` |
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Answer» Let `cos^(-1)x=theta`. Then, `x=cot(theta)`. `"cosec"theta=sqrt(1+cot^(2)theta)=sqrt(1+x^(2))` `sintheta=1/sqrt(1+x^(2))` `sin(cot^(-1)x)=1/sqrt(1+x^(2))[ therefore theta=cot^(-1)x]` `tan^(-1){sin(cot^(-1)x)}=tan^(-1)1/sqrt(1+x^(2))=phi` (say) `rArr cos[tan^(-1){sin(cot^(-1)x)]=cosphi`………………….(i) Now, `tan^(-1)1/sqrt(1+x^(2))=phi rArr tanphi=1/sqrt(1+x^(2))` `rArr secphi=sqrt(1+tan^(2)phi)= sqrt(1+1/(1+x^(2))=sqrt((2+x^(2))/(1+x^(2))` `rArr cosphi=sqrt(1+x^(2))/(2+x^(2))`...............(ii) From (i) and (ii), we get `cos[tan^(-1){sin(cot^(-1)x)}]=sqrt((x^(2)+1)/(x^(2)+2)]` |
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