Question

Prove that:-

Cos [tan^{​​​​-1}{sin(cot^{​​​​-1}​​x)}] = √1+x^{​​​​​​2}/√2+x^{​​​​​​2}

Answer 1

sin (cot-1 {cos (tan -1x)})


tan-1 x = A => tan A =x


sec A = √(1+x2) ==> cos A = 1/√(1+x2) so A = cos-1(1/√(1+x2))


sin (cot-1 {cos (tan -1x)}) = sin (cot-1 {cos (cos-1(1/√(1+x2))})


=sin (cot-1 {(1/√(1+x2))})


if cot-1 {(1/√(1+x2))} = B


{(1/√(1+x2))} = cotB ==> cosec B = {(√[(2+x2)/(1+x2)])}


sin B = {(√[(1+x2)/(2+x2)]} ==> B = sin -1({(√[(1+x2)/(2+x2)]})


sin {sin -1 ({(√[(1+x2)/(2+x2)]})} = √[(1+x2)/(2+x2)]


the answer is √[(1+x2)/(2+x2)]