`sin [cot^(-1) { cos ( tan ^(-1) x)}]`, जहाँ ` x gt 0`, का मान किसके बराबर है ?

Options

A. `sqrt((x^(2)+1)/(x^(2)+2))`
B. `sqrt((x^(2)+2)/(x^(2)+1))`
C. `(sqrt((x^(2)+1)/(x^(2)+2)))^(2)`
D. `((x^(2)+2)/(x^(2)+1))`

Answer

A

Explanation

माना ` theta = tan^(-1) x`
` :' cos theta= 1/sqrt(1+tan^(2) theta)=1/sqrt(1+x^(2))`
` rArr cos (tan^(-1)x)=1/sqrt(1+x^(2))`
तब , ` cot^(-1){cos(tan^(-1)x)} = cot^(-1){1/sqrt(1+x^(2))}`
और `sin{cot^(-1)(1/sqrt(1+x^(2)))}`
पुनः माना `cot^(-1){1/sqrt(1+x^(2))} = phi rArr cot phi = 1/sqrt(1+x^(2))`
और `sin phi = 1/sqrt(1+cot^(2) phi)=(sqrt(1+x^(2)))/sqrt(x^(2)+1+1)=sqrt((x^(2)+1)/(x^(2)+2))`
`:. sin [cot^(-1){cos(tan^(-1)x)}]= sqrt((x^(2)+1)/(x^(2)+2))`