Answer
The identity is verified.
$\frac{1}{tan~\beta}+tan~\beta$: black
$\frac{sec^2\beta}{tan~\beta}$: red
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Work Step by Step
$\frac{1}{tan~\beta}+tan~\beta=\frac{1}{tan~\beta}+tan~\beta~\frac{tan~\beta}{tan~\beta}=\frac{1+tan^2\beta}{tan~\beta}=\frac{sec^2\beta}{tan~\beta}$