Double Angle Formula for Tangent/Corollary

Theorem

Let $u = \tan \dfrac \theta 2$.

Then:

$\tan \theta = \dfrac {2 u} {1 - u^2}$

where $\tan$ denotes tangent.

Proof

From Double Angle Formula for Tangent:

$\tan 2 \theta = \dfrac {2 \tan \theta} {1 - \tan^2 \theta}$

The result follows by substituting $\dfrac \theta 2$ for $\theta$.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): half-angle formulae: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): half-angle formulae: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): half-angle formula
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Tangent-of-half-angle formulae
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Tangent-of-half-angle formulae