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

This post is dedicated to Fazenda:

Secant/Tangent Equation (no sol.)

In order to establish identity, we work on one side of the equation...

In order to establish identity, we work on one side of the equation to obtain the other one.  I think it is by convention that we begin with the left side.  But it might be just my style of working math.  So I work on the left side:

If I was working on both sides of the identity instead of following the rule of working on the left side only, I would obtain the results much faster:

Cosine/Tangent/Cotangent/Cosecant Identity

Tangent/Sine Identity

Tangent/Secant Equation