#1**+2 **

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 ]

CPhill Apr 20, 2019

#3**+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 ]

CPhill Apr 20, 2019