# The Proof of the Tangent Half-Angle Formula

In this post, we prove the following trigonometric identity:

.

**Proof**

Consider a semi-circle with “center” and diameter and radius equal to 1 unit as shown below. If we let , then by the Inscribed Angle Theorem, .

Draw perpendicular to as shown in the second figure. We can compute for the sine and cosine of which equal to the lengths of and , respectively. In effect, and .

Draw . Notice that and are similar triangles, so their corresponding angles are congruent. So, .

Now, we compute for the tangent of .

In triangle ,

.

In triangle ,

.

Therefore,

.

And we are done.

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*The last figure is the proof without words of R. J. Walker.*