Question

Prove that: `"sin"[cot^(-1){"cos"(tan^(-1)x)}]=sqrt((x^2+1)/(x^2+2))`` ``cos"[tan^(-1){"sin"(cot^(-1)x)}]=sqrt((x^2+1)/(x^2+2))``

Answer 1

LHS= `sin{cot^-1 { cos(tan^-1x)}}`
let `tan^-1 x = theta`
`tan theta = x`
`cos theta = 1/(sqrt(1+x^2))`
now, `sin^-1[ cot^-1 (1/sqrt(1+x^2))]`
`= siny`
`= sqrt(x^2 + 1)/sqrt(x^2 + 2) = `RHS
2) `cos[tan^-1{sin(cot^-1 x)}] `
`cos[tan^-1{sin theta}] = cos[ tan^-1( 1/sqrt(x^2+1))]`
`= cos y`
`sqrt(x^2 + 1)/sqrt(x^2+2) =RHS`
hence proved