Find the cosine of a half-angle knowing the tangent of the angle

This post is dedicated to Fazenda:

Cos/Sin(x/2)

Cos / (1+Sin)

Cosine/Tangent/Cotangent/Cosecant Identity

Cosine/Sine Trig Equation Quadratic in Form

* use the trigonometric one
** recognize a quadratic equation
*** define an auxiliary variable t
**** after solving for t, go back to its trigonometric equivalent
***** find corresponding angles on a graph of a relevant trigonometric function

Cotangent/Cosine Equation