Tangent half-angle formula

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle. Among these are the following

From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles:

Use double-angle formulae and sin²α + cos²α = 1,

then

Q.E.D.

Also, using the angle addition and subtraction formulae for both the sine and cosine one obtains:

Pairwise addition of the above four formulae yields:

Setting ${\displaystyle a={\frac {p+q}{2}}}$and ${\displaystyle b={\frac {p-q}{2}}}$ and substituting yields:

Dividing the sum of sines by the sum of cosines one arrives at:

Applying the formulae derived above to the rhombus figure on the right, it is readily shown that

In the unit circle, application of the above shows that ${\displaystyle t=\tan \left({\frac {\varphi }{2}}\right)}$. According to similar triangles,

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