One Plus Tangent Half Angle over One Minus Tangent Half Angle

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Theorem

$\dfrac {1 + \tan \frac x 2} {1 - \tan \frac x 2} = \sec x + \tan x$


Proof

\(\ds \frac {1 + \tan \frac x 2}{1 - \tan \frac x 2}\) \(=\) \(\ds \frac {1 + \frac {\sin \frac x 2}{\cos \frac x 2} }{1 - \frac {\sin \frac x 2}{\cos \frac x 2} }\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {\cos \frac x 2 + \sin \frac x 2}{\cos \frac x 2 - \sin \frac x 2}\) multiply top and bottom by $\cos \dfrac x 2$
\(\ds \) \(=\) \(\ds \frac {\cos \frac x 2 + \sin \frac x 2}{\cos \frac x 2 - \sin \frac x 2} \times \frac {\cos \frac x 2 + \sin \frac x 2}{\cos \frac x 2 + \sin \frac x 2}\)
\(\ds \) \(=\) \(\ds \frac{\cos^2 \frac x 2 + \sin^2 \frac x 2 + 2 \cos \frac x 2 \sin \frac x 2}{\cos^2 \frac x 2 - \sin^2 \frac x 2}\)
\(\ds \) \(=\) \(\ds \frac{1 + 2 \cos \frac x 2 \sin \frac x 2}{\cos^2 \frac x 2 - \sin^2 \frac x 2}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \frac{1 + \sin x}{\cos x}\) Double Angle Formula for Sine, Double Angle Formula for Cosine
\(\ds \) \(=\) \(\ds \sec x + \tan x\) Sum of Secant and Tangent

$\blacksquare$