What is the formula for tan(2x)?

The formula for tan(2x) is derived from the double angle identities in trigonometry. For any angle, the tangent of double that angle, or tan(2x), can be expressed in terms of the tangent of the original angle x. The derivation utilizes the identity for the tangent of the sum of two angles:

[

\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}

]

When we set both a and b to x, we get:

[

\tan(2x) = \tan(x + x) = \frac{\tan x + \tan x}{1 - \tan x \tan x} = \frac{2\tan x}{1 - \tan^2 x}

]

Thus, the formula for tan(2x) is precisely:

[

\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}

]

This identity represents how the tangent function behaves under the operation of doubling an angle. It shows a relationship between the tangent of the double angle and the tangent of the original angle, making it an essential formula in trigonometric calculus