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))``

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

1.	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» 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

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