Cot2x Formula : Proof, Graph, Properties, and Examples

What Is Cot2x?

Cot2x is the cotangent of a doubled angle. Cotangent itself is the reciprocal of tangent, $\cot x = \dfrac{\cos x}{\sin x} = \dfrac{1}{\tan x}$, one of the reciprocal identities, so $\cot 2x$ measures $\dfrac{\cos 2x}{\sin 2x}$. The cot2x formula expresses that value using only $\cot x$:

$$\cot 2x = \frac{\cot^2 x - 1}{2\cot x}$$

Here $\cot^2 x$ means $(\cot x)^2$, and $x$ is any angle for which $\cot x$ and $\cot 2x$ are both defined. The identity has three other forms worth knowing, each handy in a different situation:

• In terms of $\tan x$: $\cot 2x = \dfrac{1 - \tan^2 x}{2\tan x}$

• As reciprocal of tan2x: $\cot 2x = \dfrac{1}{\tan 2x}$

• As a difference: $\cot 2x = \dfrac{1}{2}\left(\cot x - \tan x\right)$

All four describe the same curve; they belong to the wider set of trigonometric identities a student meets in the double-angle chapter.

The Rule Behind A Doubled Sighting Angle

A surveyor who doubles a sighting angle to reach a far tower leans on the cot2x rule. Cotangent turns an angle of elevation into a horizontal distance, and when two readings combine into a doubled angle, the double-angle identity for cotangent is what keeps the arithmetic exact instead of approximate.

The cot2x formula does one job: it rewrites $\cot(2x)$, the cotangent of a doubled angle, using only the cotangent of the single angle $x$. That conversion belongs to the same double-angle family as the tan2x formula, and it earns its place because cotangent problems often reach an answer with fewer steps than their tangent equivalents.

How Do You Prove The Cot2x Formula?

The cleanest proof starts from the cotangent angle-sum rule. Before using it, recall that the reference angle for any angle is the acute angle it makes with the horizontal axis — it fixes the size of a ratio, while the quadrant fixes the sign. With that in hand, the cotangent addition identity is:

$$\cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}$$

Set $A = B = x$, so that $A + B = 2x$:

$$\cot 2x = \frac{\cot x \cot x - 1}{\cot x + \cot x}$$

Simplify the numerator and denominator, one step per line:

$$\cot 2x = \frac{\cot^2 x - 1}{2\cot x}$$

That is the formula. To reach the $\tan x$ version, divide top and bottom by $\cot^2 x$ and use $\cot x = \frac{1}{\tan x}$; to reach the difference form, split the single fraction:

$$\cot 2x = \frac{\cot^2 x}{2\cot x} - \frac{1}{2\cot x} = \frac{1}{2}\cot x - \frac{1}{2}\tan x$$

What Are The Properties Of Cot2x?

The graph and the formula fix every property of $y = \cot 2x$.

• Domain: all real numbers except where $\sin 2x = 0$, that is $x = \dfrac{n\pi}{2}$, where the cotangent is undefined.

• Range: all real numbers. Unlike sine and cosine, cotangent is unbounded.

• Period: $\dfrac{\pi}{2}$. Cotangent has period $\pi$, and doubling the angle halves it.

• Odd function: $\cot(-2x) = -\cot 2x$, so the graph has rotational symmetry about the origin.

• Vertical asymptotes: at $x = \dfrac{n\pi}{2}$, where the denominator vanishes.

• Zeros: wherever $\cos 2x = 0$, that is $x = \dfrac{\pi}{4} + \dfrac{n\pi}{2}$.

Examples Of Cot2x

Example 1

Write $\cot 2x$ in terms of $\tan x$, given $\tan x = \frac{1}{2}$.

$$\cot 2x = \frac{1 - \tan^2 x}{2\tan x} = \frac{1 - \frac{1}{4}}{2 \cdot \frac{1}{2}} = \frac{\frac{3}{4}}{1} = \frac{3}{4}$$

Final answer: $\cot 2x = \dfrac{3}{4}$.

Example 2

Find $\cot 60^\circ$ using $x = 30^\circ$. First instinct, then the correct route.

The rushed move is to double the cotangent directly: $\cot 60^\circ = 2\cot 30^\circ = 2\sqrt{3} \approx 3.46$.

Check that. The cotangent decreases as the angle grows from $0^\circ$ to $90^\circ$, so $\cot 60^\circ$ must be smaller than $\cot 30^\circ = \sqrt{3} \approx 1.73$, not larger. An answer of $3.46$ moves the wrong way, so doubling the cotangent is wrong.

Use the formula instead, with $\cot 30^\circ = \sqrt{3}$:

$$\cot 60^\circ = \frac{(\sqrt{3})^2 - 1}{2\sqrt{3}} = \frac{3 - 1}{2\sqrt{3}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}}$$

Final answer: $\cot 60^\circ = \dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3}$, which matches the standard value.

Example 3

Prove $\cot 2x = \dfrac{1}{2}(\cot x - \tan x)$.

Start from the main formula and split the fraction:

$$\cot 2x = \frac{\cot^2 x - 1}{2\cot x} = \frac{\cot^2 x}{2\cot x} - \frac{1}{2\cot x}$$

$$= \frac{1}{2}\cot x - \frac{1}{2}\tan x$$

Final answer: the identity holds, since $\dfrac{1}{\cot x} = \tan x$.

Example 4

Given $\cot x = 3$, find $\cot 2x$.

$$\cot 2x = \frac{\cot^2 x - 1}{2\cot x} = \frac{9 - 1}{2 \cdot 3} = \frac{8}{6} = \frac{4}{3}$$

Final answer: $\cot 2x = \dfrac{4}{3}$.

Example 5

Evaluate $\cot 2x$ at $x = 45^\circ$ two ways.

By the $\dfrac{\cos 2x}{\sin 2x}$ form: $\cot 90^\circ = \dfrac{\cos 90^\circ}{\sin 90^\circ} = \dfrac{0}{1} = 0$.

By the main formula, with $\cot 45^\circ = 1$:

$$\cot 90^\circ = \frac{1^2 - 1}{2 \cdot 1} = \frac{0}{2} = 0$$

Final answer: $\cot 90^\circ = 0$; both routes agree.

Example 6

For $x$ in $[0, \pi]$, where is $\cot 2x$ undefined?

Cotangent is undefined where its sine denominator is zero, so $\sin 2x = 0$:

$$2x = 0, ; \pi, ; 2\pi \quad\Rightarrow\quad x = 0, ; \frac{\pi}{2}, ; \pi$$

Final answer: $\cot 2x$ is undefined at $x = 0, \dfrac{\pi}{2}, \pi$, the vertical asymptotes on that interval.

Where Does Cot2x Earn Its Keep? - "Distance from a doubled elevation angle"

The cot2x formula exists so that a doubled angle can be handled with the same exactness as the original. Cotangent is the ratio that carries an angle straight to a length.

• Surveying and heights. Cotangent converts an angle of elevation into a horizontal distance, since $\cot\theta = \dfrac{\text{adjacent}}{\text{opposite}}$. When a second reading doubles the reference angle, the double-angle identity keeps the combined distance exact.

• AC circuits and phase. The cotangent of a phase angle appears in electrical reactance calculations; doubling a driving frequency doubles the relevant angle, and cot2x expresses the new phase from the old one.

• Simplifying tangent-heavy expressions. Because $\cot 2x = \frac{1}{2}(\cot x - \tan x)$, a messy combination of $\cot x$ and $\tan x$ often collapses into a single cot2x term.

What most explainers skip is why cotangent has vertical asymptotes and sine does not. Cotangent is a ratio with $\sin 2x$ in the denominator, so wherever the doubled angle points straight up or along the axis and $\sin 2x$ hits zero, the ratio runs off to infinity. That is not a flaw in the formula - it is the honest statement that a purely horizontal or vertical sightline has no finite cotangent.

Common Mistakes With Cot2x

Mistake 1: Writing Cot2x As 2cot X

Where it slips in: Whenever the angle is doubled and the reader copies the "$2x$" straight into a coefficient.

Don't do this: $\cot 2x = 2\cot x$. Cotangent, like every trig function, is not linear.

The correct way: Use $\cot 2x = \dfrac{\cot^2 x - 1}{2\cot x}$. The direction test catches the error fast: on $(0^\circ, 90^\circ)$ cotangent falls as the angle grows, so a doubled angle gives a smaller cotangent, never a doubled one. The rusher who skips this check gets the magnitude and often the trend wrong.

Mistake 2: Confusing Cot2x With Cot²x

Where it slips in: Reading a formula sheet quickly, where $\cot 2x$ and $\cot^2 x$ sit close together.

Don't do this: Substituting $\cot^2 x$ (cotangent squared) where the problem means $\cot 2x$ (cotangent of a doubled angle).

The correct way: Keep the operations ordered. $\cot 2x$ doubles the angle first; $\cot^2 x$ squares the cotangent. They only coincide by accident at isolated angles. The second-guesser who rewrites a correct $\cot 2x$ as $\cot^2 x$ mid-solution loses the thread here.

Mistake 3: Ignoring The Undefined Points

Where it slips in: Evaluating or graphing $\cot 2x$ near $x = \dfrac{n\pi}{2}$.

Don't do this: Plugging $x = \dfrac{\pi}{2}$ into $\cot 2x$ and reporting a finite number.

The correct way: $\cot 2x$ is undefined wherever $\sin 2x = 0$. Skipping that check is the same class of error a surveyor would make by trying to read a horizontal distance from a sightline aimed straight up - the right-angled triangle simply has no finite adjacent-over-opposite ratio there, and the arithmetic returns a division by zero.

Key Takeaways

• The cot2x formula is $\cot 2x = \dfrac{\cot^2 x - 1}{2\cot x}$, the double-angle identity for cotangent.

• It is proved from the cotangent angle-sum rule with $A = B = x$.

• Equivalent forms are $\dfrac{1 - \tan^2 x}{2\tan x}$, $\dfrac{1}{\tan 2x}$, and $\dfrac{1}{2}(\cot x - \tan x)$.

• The graph has period $\dfrac{\pi}{2}$, unbounded range, and vertical asymptotes at $x = \dfrac{n\pi}{2}$.

• Cot2x is not $2\cot x$, and it is undefined wherever $\sin 2x = 0$.

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A Practical Next Step

Practice these to solidify your understanding: prove the $\frac{1}{2}(\cot x - \tan x)$ form from the main formula, then evaluate $\cot 2x$ at $x = 30^\circ$ and check it against $\cot 60^\circ$. If the asymptotes trip you up, come back to the properties section above. Want a live Bhanzu trainer to work the double-angle family with you? Book a free demo class.

Read More

• Cos 2x — the double-angle identity that feeds cot2x through cos2x/sin2x.

• Cos3x — the triple-angle cousin, built from the same angle-sum idea.

• Tan3x — the triple-angle identity for tangent.

• Double angle formula — the sin, cos, and tan doubling rules in one place.

• Trigonometric ratios — the sine, cosine, tangent, and cotangent definitions underneath.