sin (cot^{-1} {cos (tan ^{-1}x)})
tan^{-1} x = A => tan A =x
sec A = √(1+x^{2}) ==> cos A = 1/√(1+x^{2}) so A = cos^{-1}(1/√(1+x^{2}))
sin (cot^{-1} {cos (tan ^{-1}x)}) = sin (cot^{-1} {cos (cos^{-1}(1/√(1+x^{2}))})
=sin (cot^{-1} {(1/√(1+x^{2}))})
if cot^{-1} {(1/√(1+x^{2}))} = B
{(1/√(1+x^{2}))} = cotB ==> cosec B = {(√[(2+x^{2})/(1+x^{2})])}
sin B = {(√[(1+x^{2})/(2+x^{2})]} ==> B = sin ^{-1} ({(√[(1+x^{2})/(2+x^{2})]})
sin {sin ^{-1} ({(√[(1+x^{2})/(2+x^{2})]})} = √[(1+x^{2})/(2+x^{2})]
the answer is √[(1+x^{2})/(2+x^{2})]