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

This post is dedicated to Fazenda:

Evaluate the sine of -11π/12

Proved in three lines

Here is an example of a trig identity that may look challenging but is actually proved in three lines:

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:

1 + Sec Sin

Cos/Sin(x/2)

Cos / (1+Sin)

Cosine/Tangent/Cotangent/Cosecant Identity

Tangent/Sine Identity