The angle 0 whose cosine equals to its tangent is given by

Let $\theta$ be the required angle. Then, 

$\begin{array}{l}\cos \theta =\tan \theta \\ \Rightarrow {\cos }^2\theta =\sin \theta \\ \Rightarrow {\sin }^2\theta +\sin \theta -1=0\Rightarrow \sin \theta =\frac{\sqrt{5}-1}{2}=2\sin {18}^{\circ }\end{array}$