Half Angle Formula for 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
\(\ds \tan \frac \theta 2\) | \(=\) | \(\ds \frac {1 - \cos \theta} {\sin \theta}\) | Half Angle Formula for Tangent: Corollary $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sin \theta} - \frac {\cos \theta} {\sin \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \csc \theta - \cot \theta\) | 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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.43$