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# Tan2x Formula — 2tanx/(1−tan²x), Proof, Examples

TL;DR

The tan2x formula is \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \) — the double-angle identity for tangent, expressing the tangent of a doubled angle in terms of \( \tan x \) alone. This article gives the formula, two derivations (angle-addition and the \( \sin 2x / \cos 2x \) route), six worked examples in degrees and radians, the angles where it is undefined, and the mistakes that cost marks.

## What Is the Tan2x Formula?

The tan2x formula is the double-angle identity for the tangent function. It expresses \( \tan 2x \) — the tangent of twice an angle — purely in terms of \( \tan x \):  
\[ \tan 2x = \frac{2\tan x}{1 - \tan^2 x}. \]

Two equivalent forms are worth keeping nearby. From the ratio of the sine and cosine double-angle identities:
\[ \tan 2x = \frac{\sin 2x}{\cos 2x} = \frac{2\sin x \cos x}{\cos^2 x - \sin^2 x}. \]

Note the denominator \( 1 - \tan^2 x \). Whenever \( \tan x = \pm 1 \), which happens at \( x = 45°,135°,… \), the denominator is zero and \( \tan 2x \) is undefined. The formula diagnoses its own asymptotes.

## How Is the Tan2x Formula Derived?

There are two clean routes, and both are worth seeing — they reinforce each other.

**Method 1 — from the angle-addition identity.**  Write \( 2x = x + x  \) and apply the tangent sum formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}. \]
Using \( A = B = x \):  
\[ \tan 2x \ = \tan(x+x) = \frac{\tan x + \tan x}{1 - \tan x \cdot \tan x} = \frac{2\tan x}{1 - \tan^2 x}. \]

**Method 2 — from \( \sin 2x \) and \( \cos 2x \).**  Start from \( \tan 2x = \frac{\sin 2x}{\cos 2x} \), substitute the double-angle identities, then divide top and bottom by \( \cos 2x \):
\[ \tan 2x = \frac{2\sin x \cos x}{\cos^2 x - \sin^2 x} = \frac{\frac{2\sin x \cos x}{\cos^2 x}}{\frac{\cos^2 x - \sin^2 x}{\cos^2 x}} = \frac{2\frac{\sin x}{\cos x}}{1 - \frac{\sin^2 x}{\cos^2 x}} = \frac{2\tan x}{1 - \tan^2 x}. \]

Both methods land on the same identity. Method 1 is faster; Method 2 shows _why_ the denominator is \( 1 - \tan^2 x \).

## Where Is the Tan2x Formula Undefined?

\( \tan 2x \) is undefined in two distinct ways, and a careful student tracks both.

- **Denominator zero:** when \( 1 - \tan^2 x = 0 \), i.e. \( \tan x = \pm 1. \) This occurs at \( x = 45°, 135°, 225°, … \) (or \( x = \frac{\pi}{4} + k\frac{\pi}{2} \)). Here the _formula_ fails even though \( 2x \) is a perfectly ordinary angle.

- **Tangent itself undefined:** when \( \cos 2x = 0 \), i.e. \( 2x = 90°, 270°, … \), so \( x = 45°, 135°, … \). At these points \( \tan 2x \) has a genuine vertical asymptote.

The period of \( \tan 2x \) is \( \frac{\pi}{2} \) (or 90°) — half the period of \( \tan x \).

## Examples of the Tan2x Formula

### Example 1

**Given \( \tan x = \frac{3}{4}, \) find \( \tan 2x.**

Substitute directly into the formula:  
\[ \tan 2x = \frac{2 \cdot \frac{3}{4}}{1 - \left(\frac{3}{4}\right)^2} = \frac{\frac{3}{2}}{1 - \frac{9}{16}} = \frac{\frac{3}{2}}{\frac{7}{16}} = \frac{3}{2}\cdot\frac{16}{7} = \frac{24}{7}. \]
**Final answer:** \( \tan 2x = \frac{24}{7}. \)

### Example 2

**Find \( \tan 2x \) when \( \sin x = \frac{4}{5} \) and x is acute.**  
With \( \cos x = \frac{3}{5} \) from the Pythagorean theorem, \( \tan x = \frac{4}{3}. \)  Hence,  
\[ \tan 2x = \frac{2(\frac{4}{3})}{1 - (\frac{4}{3})^2} = \frac{\frac{8}{3}}{1 - \frac{16}{9}} = \frac{\frac{8}{3}}{-\frac{7}{9}} = -\frac{24}{7}. \]
**Final answer:** \( \tan 2x = -\frac{24}{7}. \)

### Example 3

**Find \( \tan 2x \) when x = 30° (π/6).**  
Here \( \tan 30° = \frac{1}{\sqrt{3}} \), and \( 2x = 60° \), so we can check against the known value.  
\[ \tan 60° = \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}. \]  
**Final answer:** \( \tan 60° = \sqrt{3}. \)

### Example 4

**Given \( \cos x = \frac{12}{13} \) and x is acute, find \( \tan 2x.**

First find \( \tan x. \) With \( \, \sin x = \frac{5}{13}, \tan x = \frac{5}{12}. \)
Then:  
\[ \tan 2x = \frac{2 \cdot \frac{5}{12}}{1 - \left(\frac{5}{12}\right)^2} = \frac{\frac{5}{6}}{1 - \frac{25}{144}} = \frac{\frac{5}{6}}{\frac{119}{144}} = \frac{120}{119}. \]
**Final answer:** \( \tan 2x = \frac{120}{119}. \)

### Example 5

**Show why \( \tan 2x \) is undefined at x = 45° (π/4).**  
At x = 45°, \( \tan 45° = 1, \) so the denominator is zero.  
\[ \tan(2 \cdot 45°) = \frac{2(1)}{1 - 1} = \frac{2}{0}; \text{ undefined}. \]
**Final answer:** undefined.

### Example 6

**Express \( \tan 2x \) purely in terms of \( \sin x, \) given \( \sin x = s \) and x is acute.**  
With \( \cos x = \sqrt{1 - s^2}, \tan x = \frac{s}{\sqrt{1 - s^2}}. \) Substitute:
\[ \tan 2x = \frac{2s}{1 - 2s^2}. \]
**Final answer:** \( \tan 2x = \frac{2s}{1 - 2s^2}. \)

## Where the Tan2x Formula Carries Real Weight

The double-angle tangent identity shows up in various applications:

- **Calculus integration:** The tangent half-angle substitution transforms trig integrals into rational ones.
- **Optics and reflection:** When a mirror rotates by an angle, the reflected ray rotates by twice that angle.
- **Surveying and slope geometry:** Computing angles uses the same identity — the slope of a line is \( \tan x \).
- **Signal processing:** Frequency-doubling in systems produces terms governed by double-angle identities.

## Tripping Points to Avoid

### Mistake 1: Dropping the denominator
**Where it slips in:**  Writing \( \tan 2x = 2 \tan x. \)
**Don't do this:** The formula relies on the denominator \( 1 - \tan^2 x. \)

### Mistake 2: Confusing \( \tan 2x \) with \( \tan^2 x \)
**Where it slips in:** Treating them as interchangeable.

### Mistake 3: Forgetting the formula can be undefined
**Where it slips in:** Plugging in values that cause a zero denominator.

## Conclusion

The \( \tan 2x \) formula is \( \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \) — the double-angle identity for tangent.  It derives from the tangent sum formula with \( A = B = x \) or from dividing \( \sin 2x \) by \( \cos 2x. \)  The two most common errors arise from misunderstanding the denominator and confusing notations.