Tangent of 15 Degrees/Proof 2

Theorem

$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$

Proof

\(\ds \tan 15 \degrees\)

\(=\)

\(\ds \tan \frac {30 \degrees} 2\)

\(\ds \)

\(=\)

\(\ds \frac {1 - \cos 30 \degrees} {\sin 30 \degrees}\)

Half Angle Formula for Tangent: Corollary $2$

\(\ds \)

\(=\)

\(\ds \frac {1 - \frac {\sqrt 3} 2} {\frac 1 2}\)

Cosine of $30 \degrees$ and Sine of $30 \degrees$

\(\ds \)

\(=\)

\(\ds 2 - \sqrt 3\)

multiplying top and bottom by $2$

$\blacksquare$