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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.