Write The Trigonometric Expression as an Algebraic Expression in u

tan^-1 (u)   = θ   which implies that

tan (θ)  =  u   =  u / 1

So.....tan =  y / x     which implies that    y = u   and  x  = 1

And   cos ( θ)  =  x / r

So    r  = √ [ x^2 +  y^2 ]  =  √[ 1^2 + u^2 ]  = √ [ 1 + u^2 ]

So....putting all this together

cos ( tan^-1 (u) )  =  cos (θ)   =  x / r  =     1 / √ [ 1 + u^2 ]

I thnk that you might mean this????

tan^-1 ( u)  = θ    means that   tan (θ)  = u  =  u / 1   =  y / x

So....by the Pythagorean Theorem.....

x^2 + y^2  =  r^2

1^2 + u^2  = r^2     take both roots

±√[ 1 + u^2 ]  =  r

So   cos  θ  =    x / r  =   ±1 / √[ 1 + u^2 ]

[ I forgot to take both roots in my first answer ]