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# Cos2x - Formula, Identity, Examples, Proof

[Trigonometry](/content/tag/trigonometry/index.html)

TL;DR

The **cos2x identity** is the double-angle formula for cosine, with three equivalent forms: \( \cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x \). The proof from the angle-sum identity, when to use each form, the related \( \cos 2x \) in terms of \( \tan x \), worked examples, and the most common student mistakes.

BT

[Bhanzu Team](/content/authors/bhanzu-team/index.html) Last updated on May 15, 2026 8 min read

## What Is the Cos2x Formula?

The **cos2x formula** is the double-angle identity for cosine. It expresses \( \cos \) of \( 2x \) — twice an angle — in terms of \( \cos x \) and \( \sin x \) alone.

Three equivalent forms (every form is exactly equal — pick the one that fits your problem):

1. \( \cos 2x = \cos^2 x - \sin^2 x \quad \text{(Form 1)} \)
2. \( \cos 2x = 2\cos^2 x - 1 \quad \text{(Form 2)} \)
3. \( \cos 2x = 1 - 2\sin^2 x \quad \text{(Form 3)} \)

There is also a fourth form in terms of tan:

4. \( \cos 2x = \frac{1 - \tan^2 x}{1 + \tan^2 x} \quad \text{(Form 4)} \)

## How Do You Prove Cos2x = Cos²x − Sin²x?

Start from the sum formula for cosine:

\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Set \( A = B = x \):

\( \cos(x + x) = \cos x \cos x - \sin x \sin x \)

\( \cos 2x = \cos^2 x - \sin^2 x \)

**That's Form 1, derived in two lines.**

## How Do You Get the Other Two Forms?

Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \).

### Form 2: Cos2x in Terms of Cos Only

Replace \( \sin^2 x \) with \( 1 - \cos^2 x \):

\( \cos 2x = \cos^2 x - (1 - \cos^2 x) = 2\cos^2 x - 1 \)

### Form 3: Cos2x in Terms of Sin Only

Replace \( \cos^2 x \) with \( 1 - \sin^2 x \):

\( \cos 2x = (1 - \sin^2 x) - \sin^2 x = 1 - 2\sin^2 x \)

### Form 4: Cos2x in Terms of Tan

Divide Form 1's numerator and denominator (1 in the denominator) by \( \cos^2 x \):

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

## Is Cos2x the Same as Cos²x?

**No.** \( \cos 2x \) and \( \cos^2 x \) look similar on the page but are entirely different functions. Confusing them is the single most common notation mistake in trigonometry.

| Notation          | Meaning                                      | Worked at x=30°           |
|-------------------|----------------------------------------------|----------------------------|
| \( \cos 2x \)   | Cosine _of_ the angle 2x                    | \( \cos 60° = \frac{1}{2} \)  |
| \( \cos^2 x \) | Cosine of x, squared: \( (\cos x)^2 \) | \( (\cos 30°)^2 = \frac{3}{4} \) |

Two different values from the same \( x=30° \).

**How to tell them apart at a glance:**
- The "2" sits **before the variable** in \( \cos 2x \) → "double angle." It's part of the _argument_.
- The "2" sits **between cos and the variable** in \( \cos^2 x \) → "cosine squared." It's an exponent applied to the _output_.

**Worked relationship:** The cos2x identity in Form 2 connects the two:

\( \cos 2x = 2\cos^2 x - 1 \quad \Leftrightarrow \quad \cos^2 x = \frac{1 + \cos 2x}{2} \)

## When Do You Use Each Form?

| Form                             | Use When                                 |
|----------------------------------|------------------------------------------|
| \( \cos^2 x - \sin^2 x \)    | Both \( \sin x \) and \( \cos x \) are known |
| \( 2\cos^2 x - 1 \)           | Only \( \cos x \) is known             |
| \( 1 - 2\sin^2 x \)           | Only \( \sin x \) is known             |
| \( \frac{1 - \tan^2 x}{1 + \tan^2 x} \) | Only \( \tan x \) is known             |

## Worked Examples

### Example 1: Direct Computation

Compute \( \cos 60° \) using the cos2x identity with \( x=30° \).

Use Form 2 (only \( \cos 30° \) needed):

\( \cos 60° = 2\cos^2 30° - 1 = 2 \cdot \frac{3}{4} - 1 = \frac{1}{2} \)

### Example 2: When Sin is Given

If \( \sin x = \frac{3}{5} \), find \( \cos 2x \).

Use Form 3 (only \( \sin x \) needed):

\( \cos 2x = 1 - 2\sin^2 x = 1 - 2 \cdot \frac{9}{25} = \frac{7}{25} \)

### Example 3: Solving a Trigonometric Equation

Solve \( \cos 2x + \cos x = 0 \) for \( x \in [0, 2\pi) \).

Let \( u = \cos x \):

\( 2u^2 + u - 1 = 0 \)

Factoring: \( (2u−1)(u+1)=0 \)

Solutions: \( u=\frac{1}{2}, u=-1 \) gives \( x=\frac{\pi}{3},\pi,\frac{5\pi}{3} \).

## Why Does Cos2x Matter?

> _"The double-angle formula is the gateway to power-reduction in calculus."_ — adapted from any standard calculus text.

The cos2x identity isn't just an academic identity. It plays a significant role in several practical computations:
- **Integration in calculus.** The integral of \( \cos^2 x \) is simplified using the identity: \( \int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C \).
- **Signal processing.** Used in JPEG and MP3 encoding through cosine transformations.
- **Physics — power in AC circuits.** Average power calculation uses the identity.

The cos2x identity is attributed to Leonhard Euler's work in the 18th century, but similar concepts were utilized centuries earlier by Indian mathematicians like Bhaskara II in their trigonometric work.

## A Worked Example

Find \( \cos 2x \) given \( \sin x = \frac{4}{5} \) with \( x \) in Q2.

Use Form 3:

\( \cos 2x = 1 - 2\sin^2 x = 1 - 2 \cdot \frac{16}{25} = -\frac{7}{25} \)

Check: \( \cos 2x = -\frac{7}{25} \).

## What Are the Most Common Mistakes With Cos2x?

### **Mistake 1: Treating cos2x as 2cosx**
**Don't do this:** Treat \( \cos 2x \) as twice \( \cos x \).
### **Mistake 2: Forgetting to apply the correct sign in Form 2**
**Where it slips in:** Seems easy to recall, but pay attention to the negative sign.
### **Mistake 3: Picking the slowest form for the problem**
**Don't do this:** Always Matching the form to the information available saves work.

## The Mathematicians Who Shaped the Cos2x Identity
> **Leonhard Euler (1707–1783)** — Standardised modern double-angle identities in mathematics.
> **Bhaskara II (1114–1185)** — Used double-angle relationships in his work on trigonometry for astronomy.

## A Practical Next Step
Try these exercises to reinforce your understanding before proceeding to triple-angle formulas.