$$\begin{align}\sec\theta \;-\; \tan\theta &\;=\; \tan\phi \;=\; \tan\left(\;45^\circ \;-\; \frac{\theta}{2}\;\right) \\[6pt]
\sec\theta \;+\; \tan\theta &\;=\; \tan\psi \;=\; \tan\left(\;45^\circ \;+\; \frac{\theta}{2}\;\right)\end{align}$$
\sec\theta \;+\; \tan\theta &\;=\; \tan\psi \;=\; \tan\left(\;45^\circ \;+\; \frac{\theta}{2}\;\right)\end{align}$$
Formula name inspired by Michael Hardy at Math.StackExchange.