Cartographer’s Tangent Formulas

trigonograph-cartographerstangent

$$\large \begin{align}\sec\theta \;-\; \tan\theta &\;=\; \tan\phi \;=\; \tan\left(\;45^\circ \;-\; \frac{\theta}{2}\;\right) \\[6pt]
\large \sec\theta \;+\; \tan\theta &\;=\; \tan\psi \;=\; \tan\left(\;45^\circ \;+\; \frac{\theta}{2}\;\right)\end{align}$$

Formula name inspired by Michael Hardy at Math.StackExchange.