What are the double angle identities?

The double angle identities express sin(2x), cos(2x) and tan(2x) in terms of sin x, cos x and tan x. sin(2x) = 2 sin x cos x. cos(2x) = cos squared x minus sin squared x, which is also equal to 2 cos squared x minus 1, and to 1 minus 2 sin squared x. tan(2x) = 2 tan x divided by (1 minus tan squared x). They all follow from the angle sum identities sin(A + B) and cos(A + B) with A and B both set to x.

How do you check a double angle identity is correct?

Pick a test angle, work out sin x, cos x and tan x, then calculate 2x directly (for example using a calculator's sin, cos and tan buttons on the doubled angle) and compare it with the value produced by the identity. If both methods give the same number, the identity holds for that angle. Because these identities are true for every real value of x, testing several different angles (including negative angles and angles past 90 degrees) is a reliable way to confirm you have applied the identity correctly.

Why do the three cos(2x) forms always agree?

All three forms come from the same identity, cos(2x) = cos squared x minus sin squared x, rewritten using the Pythagorean identity sin squared x plus cos squared x = 1. Because sin squared x and cos squared x are never independent (one is always 1 minus the other), substituting one for the other cannot change the final numeric value. If a calculation shows the three forms disagreeing, it means there is a rounding or input error rather than a flaw in the identity itself.

Double Angle Identities Calculator

Enter an angle x and this calculator checks the double angle identities for you: sin(2x), all three equivalent forms of cos(2x) tested against one another, and tan(2x). It confirms whether the identities agree for your chosen angle and shows every step of the check.

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Reference formula Standard trigonometric double angle identities, valid for all real values of x.

1. Angle

Angle x

Angle unit

2. Display

Decimal places

Identity Check for 2x

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

cos(2x)
-
= cos²x − sin²x

tan(2x)
-
= 2 tan x / (1 − tan²x)

Values used

Angle x-

Doubled angle 2x-

sin x-

cos x-

tan x-

sin(2x) computed directly-

cos(2x) identity check: all three forms

Form 1: cos²x − sin²x-

Form 2: 2cos²x − 1-

Form 3: 1 − 2sin²x-

cos(2x) computed directly-

Identity check result-

Summary: Enter an angle above.

What the double angle identities say

The double angle identities let you express the sine, cosine or tangent of a doubled angle (2x) in terms of the sine, cosine and tangent of the original angle (x). They are identities, meaning they hold true for every real value of x, not just for specific angles. This calculator checks that fact for whatever angle you enter, by working out both sides of each identity independently and comparing them.

sin(2x) = 2 sin x cos x
cos(2x) = cos²x − sin²x = 2cos²x − 1 = 1 − 2sin²x
tan(2x) = 2 tan x / (1 − tan²x)

How the identity check works

For each angle you enter, the calculator computes sin(2x) and cos(2x) two independent ways: directly, by doubling the angle and taking its sine or cosine, and via the identity, by combining sin x, cos x and tan x according to the formula. If both methods agree to within a tiny rounding tolerance, the identity check passes. This is the same method used to verify any trigonometric identity: substitute numbers in for the variable and confirm both sides match.

Where these identities come from

Identity	Derived from
sin(2x) = 2 sin x cos x	sin(A + B) = sin A cos B + cos A sin B, with A = B = x
cos(2x) = cos²x − sin²x	cos(A + B) = cos A cos B − sin A sin B, with A = B = x
cos(2x) = 2cos²x − 1	Form 1, with sin²x replaced by (1 − cos²x)
cos(2x) = 1 − 2sin²x	Form 1, with cos²x replaced by (1 − sin²x)
tan(2x) = 2 tan x / (1 − tan²x)	Dividing sin(2x) by cos(2x) and dividing top and bottom by cos²x

Worked example

For x = 30 degrees: sin x = 0.5, cos x = 0.866025, tan x = 0.577350. Doubling directly, 2x = 60 degrees, so sin(2x) = 0.866025 and cos(2x) = 0.5. Checking via the identities: sin(2x) = 2 × 0.5 × 0.866025 = 0.866025, matching the direct value. cos(2x) form 1 gives 0.866025² − 0.5² = 0.75 − 0.25 = 0.5. Form 2 gives 2(0.75) − 1 = 0.5. Form 3 gives 1 − 2(0.25) = 0.5. All three match the direct value of 0.5, so the identity check passes. tan(2x) = 2(0.577350) / (1 − 0.333333) = 1.154701 / 0.666667 = 1.732051, matching tan(60°) directly. Enter 30 degrees above to see this calculator return the same figures.

Common uses

Checking working when simplifying trigonometric expressions or proving identities for NCEA and university calculus courses
Confirming power-reduction substitutions used when integrating sin²x or cos²x
Verifying doubled-frequency terms in physics problems involving waves, oscillations and AC electrical signals
Building confidence with the half angle identities, which are the double angle identities rearranged and solved for x/2

What this calculator assumes

You enter one angle x in degrees or radians
tan(2x) is undefined where cos(2x) = 0 (for example when x = 45 degrees), and the calculator shows "undefined" in that case
The identity check uses a small rounding tolerance to allow for floating point precision, so it will report "All forms match" even with tiny rounding differences in the final decimal place

Related Calculators

Maths and Stats Calculators: the full hub of maths and statistics tools.
Double Angle Calculator: a quick sin(2x), cos(2x), tan(2x) evaluator.
Half Angle Calculator: the reverse relation, for x/2.
Trig Identity Calculator: check and simplify trigonometric identities.
SOHCAHTOA Calculator: basic sine, cosine and tangent ratios.

Sources: Wolfram MathWorld, Trigonometric Addition Formulas (mathworld.wolfram.com/TrigonometricAdditionFormulas.html). Wikipedia, List of trigonometric identities (en.wikipedia.org/wiki/List_of_trigonometric_identities).

This calculator applies standard trigonometric double angle identities, which hold for all real values of x. Results are for educational and reference use.