let 

let 

Substituting:



Under the radical multiply top and bottom by the conjugate
of the denominator, as if you were rationalizing the denominator:





The numerator is a perfect square, so we take the square root of
the numerator and denominator:



Write the right side as the sum of two fractions:



Next we draw a right triangle with an angle θ, 1 as the hypotenuse, ab
as the adjacent side and the radical as the opposite side:



So now the equation is 

 where 

Since you want y as a function of cot(θ), we use an identity for csc(θ).

1 + cot2(θ) = csc2(θ), solve for csc(θ)

    

And we substitute back for ab:

The final equation is 

 where  


Edwin