Half Angle Formulas/Tangent/Corollary 3

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Theorem

$\tan \dfrac \theta 2 = \csc \theta - \cot \theta$

where $\tan$ denotes tangent, $\csc$ denotes cosecant and $\cot$ denotes cotangent.


When $\theta = k \pi$, the right hand side of this formula is undefined.


Proof

\(\displaystyle \tan \frac \theta 2\) \(=\) \(\displaystyle \frac {1 - \cos \theta} {\sin \theta}\) Half Angle Formula for Tangent: Corollary 2
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \csc \theta - \cot \theta\) Cosecant is Reciprocal of Sine and Cotangent is Cosine divided by Sine


When $\theta = k \pi$, both $\cot \theta$ and $\csc \theta$ are undefined.

$\blacksquare$


Sources