If cosA = sinA then tan A/2 is

cosA = sinA
Express sinA and cos A in terms of tan A/2
$\begin{array}{r}\frac{1-{\tan }^{2}A/2}{1+{\tan }^{2}A/2}=\frac{2\tan A/2}{1+{\tan }^{2}A/2}\\ 2\tan A/2=1-{\tan }^{2}A/2\\ {\tan }^{2}A/2+2\tan A/2=1\end{array}$
Adding 1 on both sides
$\begin{array}{r}{\tan }^{2}A/2+2\tan A/2+1=2\\ {\left({\tan }^{2}A/2+1\right)}^2=2\\ \tan A/2=\pm \left(\sqrt{2}-1\right)\end{array}$