Prove that `cos (tan^(-1) (sin (cot^(-1) x))) = sqrt((x^(2) + 1)/(x^(2) + 2))`

Explanation

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)]`