• Learning Platform
• /
• Knowledge Base
• /
• Cos2x – Formula, Identity, and Solved Math Tasks

Cos2x – Formula, Identity, and Solved Math Tasks

Jessica Kaminski

reviewed by Jo-ann Caballes

Updated on August 4, 2026

Cos2x is the cosine of a doubled angle, and it can be written in three equivalent ways: cos(2x) = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x. This is the double-angle identity for cosine, a standard tool in high school trigonometry and precalculus.

What Is Cos2x?

Cos2x is the cosine of an angle that has been doubled. “Cos” is the cosine function, and 2x means the angle x multiplied by two. For example, if x = 30°, then cos2x = cos(60°) = ½.

The Cos2x Formula

There is no single cos2x formula — there are three equivalent forms, and you choose whichever fits the information you already have.

There is also a tangent form, cos(2x) = (1 − tan²x) / (1 + tan²x), which is handy when only tan x is known.

Derivation of the Cos2x Formula

The cos2x formula comes straight from the cosine addition formula, cos(A + B) = cos A cos B − sin A sin B. Setting A = B = x gives cos(2x) = cos²x − sin²x. Substituting specifically this pythagorean identities (sin²x + cos²x = 1) into that result produces the other two forms, 2cos²x − 1 and 1 − 2sin²x.

Cos2x and Other Trigonometric Functions

Cos2x connects closely to the other trigonometric functions. Using the Pythagorean identity, the double-angle partner of cos2x is sin(2x) = 2 sin x cos x. The cosine-squared identity, cos²x = (1 + cos 2x) / 2, lets you rewrite powers of cosine — useful when simplifying expressions or evaluating integrals.

Properties of Cos2x

Cos2x is an even function: cos(2x) = cos(−2x), so it is unchanged when x is replaced by −x. As a function of x it is periodic with a period of π (180°) — doubling the angle inside the cosine halves the usual 2π period of cos x, so the graph repeats twice as often.

Solving Equations With Cos2x

Simple cos2x equations are solved by isolating 2x and then dividing by 2. For cos(2x) = 0, find the angles where cosine is zero, then halve them. More complex equations use the identities: rewriting a mix of sin x and cos x in terms of cos2x often collapses the equation into something you can factor and solve.

Solved Examples

Example 1

Find cos2x when x = 30°.

Answer: ½. Doubling gives 2x = 60°, and cos(60°) = ½.

Example 2

Evaluate cos2x when x = 45°.

Answer: 0. Doubling gives 2x = 90°, and cos(90°) = 0.

Example 3

Solve cos(2x) = cos(x) for x in the interval [0, 2π).

Answer: x = 0, 2π/3, and 4π/3. The equation holds when 2x = x + 2πn (giving x = 0) or 2x = −x + 2πn (giving x = 2πn/3). Within [0, 2π), that yields 0, 2π/3, and 4π/3; note that 2π itself is excluded from the half-open interval.

Cos2x: Practice Math Problems

Cos2x Worksheets

Put this into practice with related Brighterly worksheets:

• Trigonometry worksheets
• Area and circumference of a circle worksheets
• Angles in a triangle worksheets

Frequently Asked Questions About Cos2x

What Does Cos2x Represent?

Cos2x represents the cosine of a doubled angle. If you start with an angle x, then cos2x is the cosine of the angle that is twice as large, 2x. It is one of the double-angle identities, which express the trigonometric function of a doubled angle in terms of the original angle.

What Is the Formula for Cos2x?

There are three equivalent formulas: cos(2x) = cos²x − sin²x, cos(2x) = 2cos²x − 1, and cos(2x) = 1 − 2sin²x. All three come from the basic double-angle identity combined with the Pythagorean identity. You pick the form that matches the values you already know — sin x, cos x, or both.

How Is Cos2x Used in Real-World Applications?

Cos2x appears wherever periodic behavior is modeled. Engineers and physicists use it in signal processing, wave analysis, and the study of oscillations and vibrations, and it also simplifies certain integrals in calculus. In each case, the double-angle identity helps rewrite a complicated trigonometric expression into a simpler, solvable form.

How Can I Solve Equations Involving Cos2x?

Use algebra together with trigonometric identities. Depending on the equation, you might apply the Pythagorean identity (sin²x + cos²x = 1), the even-function property, or one of the double-angle forms of cos2x. The goal is usually to rewrite the equation in terms of a single function so you can factor it and solve for x.

Is Cos2x an Even or Odd Function?

Cos2x is an even function. This means cos(2x) = cos(−2x) for every value of x, so replacing x with −x leaves the value unchanged. This follows directly from the fact that the basic cosine function is itself even, and it can make calculations with negative angles simpler.

Is Cos2x the Same as Cos(x²)?

No, they are different. Cos(2x) means the cosine of double the angle, so you multiply the angle by 2 before taking the cosine. Cos(x²) means the cosine of x squared, so you square x first and then take the cosine of that result. The order of operations changes the value completely.

Jessica Kaminski

84 articles

Jessica is a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.