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))``

Explanation

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)}] `

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