Tan Double angle formula
Expansion form
$\tan{2\theta} \,=\, \dfrac{2\tan{\theta}}{1-\tan^2{\theta}}$
Simplified form
$\dfrac{2\tan{\theta}}{1-\tan^2{\theta}} \,=\, \tan{2\theta}$
Introduction
It is called tan double angle identity and used as a formula in two cases.
- Tan of double angle is expanded as the quotient of twice the tan function by subtraction of square of tan function from one.
- The quotient of twice the tan function by subtraction of square of tan function from one is simplified as tan of double angle.
How to use
The tangent of double angle identity is used to either expand or simplify the double angle functions like $\tan{2A}$, $\tan{2x}$, $\tan{2\alpha}$ and etc. For example,
$(1) \,\,\,\,\,\,$ $\tan{2x} \,=\, \dfrac{2\tan{x}}{1-\tan^2{x}}$
$(2) \,\,\,\,\,\,$ $\tan{2A} \,=\, \dfrac{2\tan{A}}{1-\tan^2{A}}$
$(3) \,\,\,\,\,\,$ $\tan{2\alpha} \,=\, \dfrac{2\tan{\alpha}}{1-\tan^2{\alpha}}$
Proof
Learn how to derive the rule of tan double angle identity by geometric approach in trigonometry.