let x>0. write the expression as an algebraic expression in x: csc(arctan(x))

Answer 1

Draw right triangle ABC, with sides labeled so that tanA = x/1
you can see that cscA = √(x^2+1)/x

Answer 2

If arctan ( x ) = θ

then

tan θ = x

cot θ = 1 / tan θ = 1 / x

csc² θ = 1 + cot² θ = 1+ ( 1 / x )² = 1 + 1 / x² =

x² / x² + 1 / x² = ( x² + 1 ) / x²

csc θ = √( x² + 1 ) / √x²

csc θ = √( x² + 1 ) / x

So

csc ( arctan ( x ) ) = √( x² + 1 ) / x