$$\rho=\cot\phi \csc\phi .$$
Work Step by Step
We have $$z=x^2+y^2 \Longrightarrow \rho\cos\phi =\rho^2\sin^2\phi\cos^2\theta+\rho^2\sin^2\phi\sin^2\theta\\ =\rho^2\sin^2\phi .$$ That is $$\rho=\cot\phi \csc\phi .$$
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