Tan3x - Formula, Proof & Examples (Triple Angle)

What Is Tan3x?

Tan3x is the tangent of three times an angle $x$, and the tan3x formula expresses it entirely in terms of $\tan x$:

$$\tan 3x = \dfrac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$$

It is one of the triple-angle identities, the tangent counterpart to the double-angle tan2x formula. Equivalently, since tangent is the ratio of sine to cosine, $\tan 3x = \dfrac{\sin 3x}{\cos 3x}$ — but the form above is the one used for simplification, because it needs only $\tan x$.

A quick warning that the notation invites: $\tan 3x$ is not $\tan^3 x$. The first is the tangent of the angle $3x$; the second is $(\tan x)^3$, tangent cubed. They appear together inside this very formula, so reading them apart is essential. The identity sits in the broader set of trigonometric identities.

How Do You Prove the Tan3x Formula?

The derivation splits $3x$ into $2x + x$ and applies two identities you already have. Each step sits on its own line so nothing is skipped.

Step 1 — Split the angle and apply the addition formula

Using $\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A\tan B}$ with $A = 2x$, $B = x$:

$$\tan 3x = \tan(2x + x) = \dfrac{\tan 2x + \tan x}{1 - \tan 2x,\tan x}$$

Step 2 — Substitute the double-angle formula

Replace $\tan 2x = \dfrac{2\tan x}{1 - \tan^2 x}$:

$$\tan 3x = \dfrac{\dfrac{2\tan x}{1 - \tan^2 x} + \tan x}{1 - \dfrac{2\tan x}{1 - \tan^2 x}\cdot\tan x}$$

Step 3 — Clear the inner fractions

Multiply numerator and denominator by $(1 - \tan^2 x)$. The numerator becomes:

$$2\tan x + \tan x(1 - \tan^2 x) = 2\tan x + \tan x - \tan^3 x = 3\tan x - \tan^3 x$$

The denominator becomes:

$$(1 - \tan^2 x) - 2\tan^2 x = 1 - 3\tan^2 x$$

Step 4 — Assemble

$$\boxed{\tan 3x = \dfrac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}}$$

Letting $t = \tan x$, the compact form is $\tan 3x = \dfrac{3t - t^3}{1 - 3t^2}$.

Tan3x in Calculus

Two operations come up often enough to state directly.

Derivative. By the chain rule on the angle $3x$:

$$\dfrac{d}{dx}(\tan 3x) = 3\sec^2 3x$$

This is the same chain-rule pattern as the derivative of tan 2x — the multiplier out front is whatever number multiplies $x$ inside.

Integration. Writing $\tan 3x = \dfrac{\sin 3x}{\cos 3x}$ and substituting $u = \cos 3x$:

$$\int \tan 3x , dx = -\dfrac{1}{3}\ln|\cos 3x| + C = \dfrac{1}{3}\ln|\sec 3x| + C$$

Examples of Tan3x

Example 1

Find $\tan 3x$ when $\tan x = \dfrac{1}{\sqrt{3}}$ (i.e. $x = 30°$).

Let $t = \dfrac{1}{\sqrt{3}}$, so $t^2 = \dfrac{1}{3}$ and $t^3 = \dfrac{1}{3\sqrt{3}}$.

$$\tan 3x = \dfrac{3\cdot\frac{1}{\sqrt{3}} - \frac{1}{3\sqrt{3}}}{1 - 3\cdot\frac{1}{3}} = \dfrac{\frac{3}{\sqrt{3}} - \frac{1}{3\sqrt{3}}}{1 - 1}$$

The denominator is 0, so $\tan 3x$ is undefined — consistent with $\tan 90° = \tan(3 \times 30°)$ being undefined.

Final answer: undefined (as expected, since $3x = 90°$).

Example 2

Evaluate $\tan 3x$ at $\tan x = 1$ (i.e. $x = 45°$). Wrong path first.

The tempting move is to write $\tan 3x = 3\tan x = 3$, treating the "3" as a simple multiplier.

Check it: $3x = 135°$, and $\tan 135° = -1$, not 3. So multiplying the tangent by 3 is wrong — tangent does not scale linearly with the angle.

The rescue is the formula. With $t = 1$:

$$\tan 3x = \dfrac{3(1) - (1)^3}{1 - 3(1)^2} = \dfrac{3 - 1}{1 - 3} = \dfrac{2}{-2} = -1$$

Final answer: $\tan 3x = -1$, matching $\tan 135°$.

Example 3

Compute $\tan 135°$ using the formula with $x = 45°$.

This is the same setup as Example 2 read the other direction: $t = \tan 45° = 1$.

$$\tan 135° = \dfrac{3 - 1}{1 - 3} = -1$$

Final answer: $\tan 135° = -1$.

Example 4

Prove $\tan 180° = 0$ using the formula with $x = 60°$.

$t = \tan 60° = \sqrt{3}$, so $t^2 = 3$ and $t^3 = 3\sqrt{3}$.

$$\tan 180° = \dfrac{3\sqrt{3} - 3\sqrt{3}}{1 - 3(3)} = \dfrac{0}{-8} = 0$$

Final answer: $\tan 180° = 0$.

Example 5

Show that $\tan 3x,\tan 2x,\tan x = \tan 3x - \tan 2x - \tan x$.

Start from $\tan 3x = \tan(2x + x) = \dfrac{\tan 2x + \tan x}{1 - \tan 2x\tan x}$.

Cross-multiply:

$$\tan 3x,(1 - \tan 2x\tan x) = \tan 2x + \tan x$$ $$\tan 3x - \tan 3x\tan 2x\tan x = \tan 2x + \tan x$$

Rearrange:

$$\tan 3x - \tan 2x - \tan x = \tan 3x\tan 2x\tan x$$

Final answer: identity proved.

Example 6

Find the period of $y = \tan 3x$.

The period of $\tan x$ is $\pi$. Compressing the angle by 3 divides the period by 3:

$$\text{Period} = \dfrac{\pi}{3}$$

Final answer: period $= \dfrac{\pi}{3}$.

The first-instinct error students reach for in this set is "scaling" — assuming $\tan 3x = 3\tan x$ the way $3x$ is three times $x$. The formula exists precisely because tangent is non-linear, and the habit it fixes is checking the answer against a known angle like $135°$.

Why Tan3x Matters - "Untangling tripled angles"

The tan3x formula exists to answer multiplied-angle questions without ever leaving the original variable. Many problems hand you a relationship in $\tan x$ and ask for something in $\tan 3x$; converting the angle by going through degrees would lose the algebraic link the problem depends on.

• Harmonics. A wave at three times a base frequency is a third harmonic; expressing it through the base tangent is how acoustics and electronics analyse overtones.

• Equation solving. Triple-angle identities turn an equation in $\tan 3x$ into a cubic in $\tan x$ — solvable by factoring rather than guesswork.

• The non-linearity lesson. The formula's whole point is that trigonometric functions of a multiplied angle are not the function multiplied. That single idea reshapes how a student reads every later identity.

What most references state but rarely stress: the cubic on top, $3\tan x - \tan^3 x$, is exactly why a triple-angle equation can have up to three distinct solutions per period.

Common Mistakes With Tan3x

Mistake 1: Confusing tan3x with tan³x

Where it slips in: Reading or writing the formula, where both appear.

Don't do this: Treating $\tan 3x$ (tangent of $3x$) as $\tan^3 x$ (tangent cubed).

The correct way: $\tan 3x$ is one function of a tripled angle; $\tan^3 x = (\tan x)^3$ is a cube. The numerator of the formula contains both — $3\tan x$ minus $\tan^3 x$.

The point of confusion between a multiplied angle and a raised power is the most common source of wrong answers here; the cube notation $\tan^3 x$ and the angle notation $\tan 3x$ differ only in where the 3 sits.

Mistake 2: Assuming tan3x = 3 tan x

Where it slips in: Any quick mental estimate of a tripled angle.

Don't do this: Writing $\tan 3x = 3\tan x$ because $3x$ is three times $x$.

The correct way: Use the full identity. Tangent is non-linear, so the tripled angle needs the cubic numerator and the $1 - 3\tan^2 x$ denominator.

The rusher applies linear intuition to a non-linear function; the habit the formula fixes is never letting an angle multiplier pass straight through to the function value.

Mistake 3: Dropping a sign during the expansion

Where it slips in: Step 3 of the proof, clearing the inner fractions.

Don't do this: Writing the denominator as $1 + 3\tan^2 x$ by mis-signing the $-2\tan^2 x$ term.

The correct way: $(1 - \tan^2 x) - 2\tan^2 x = 1 - 3\tan^2 x$ — both tangent-squared terms are subtracted.

Key Takeaways

• The tan3x formula is $\dfrac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$.

• It is derived from $\tan(2x + x)$ using the addition and double-angle identities.

• Its derivative is $3\sec^2 3x$; its integral is $\dfrac{1}{3}\ln|\sec 3x| + C$.

• $\tan 3x$ (triple angle) is not $\tan^3 x$ (tangent cubed).

• The period of $\tan 3x$ is $\dfrac{\pi}{3}$.

To take tan3x further with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for identity practice, or live math tutoring with peers from 20+ countries.

A Practical Next Step

Practice these to solidify your understanding: compute $\tan 3x$ for $\tan x = \dfrac{1}{2}$, then prove $\tan 3x = -1$ when $x = 45°$ directly from the formula. If you get stuck on the expansion, come back to the four-step proof. Want a live Bhanzu trainer to walk through the derivation? Book a free demo class.

Read More

• Sum and difference identities — the addition formula behind the proof.

• Double angle formula — the tan2x identity substituted in Step 2.

• Trigonometric functions — the periodic behaviour behind the π/3 period.

• Reciprocal identities — the sec, csc, cot relationships used in the proof.

• Cos2x — a companion double-angle identity.