What are the double angle formulas?

The three double angle formulas are: sin(2x) = 2 sin x cos x, cos(2x) = cos squared x minus sin squared x (which also equals 2 cos squared x minus 1, or 1 minus 2 sin squared x), and tan(2x) = 2 tan x divided by (1 minus tan squared x). Each one comes from the angle sum formula with both angles set equal to x.

Why does cos(2x) have three different versions?

The base form cos(2x) = cos squared x minus sin squared x can be rewritten using the Pythagorean identity sin squared x plus cos squared x = 1. Substituting sin squared x = 1 minus cos squared x gives cos(2x) = 2 cos squared x minus 1. Substituting cos squared x = 1 minus sin squared x gives cos(2x) = 1 minus 2 sin squared x. All three forms always give the same numeric answer, so you choose whichever is more convenient for the problem you are solving.

When is the tan(2x) formula undefined?

tan(2x) = 2 tan x / (1 minus tan squared x) is undefined whenever the denominator is zero, which happens when tan x = 1 or tan x = -1, that is at x = 45 degrees, 135 degrees, 225 degrees, 315 degrees and every 90 degree interval from there. It is also undefined whenever x itself makes tan x undefined, at x = 90 degrees plus any multiple of 180 degrees.

Double Angle Formula Calculator

This calculator works out all three double angle trigonometric identities from a single angle you enter: sin(2x), cos(2x) and tan(2x), so you do not have to derive them by hand. Type in your angle x, choose whether it is in degrees or radians, and pick how many decimal places to show, and the tool instantly returns sin(2x) and tan(2x) plus the three equivalent forms of cos(2x): cos squared x minus sin squared x, 2cos squared x minus 1, and 1 minus 2sin squared x. Alongside the headline results you get a breakdown of the working, including x, 2x, and the sine, cosine and tangent of the original angle, plus a check confirming all three cos(2x) forms agree, since they always should for any real angle. This makes the calculator useful for checking homework and exam working in NCEA and university trigonometry, for simplifying expressions before integrating terms like sine squared x or cosine squared x, and for deriving related identities such as the half angle formulas. Watch for tan(2x): it is undefined whenever cos(2x) equals zero, which happens at 45 degrees and every 90 degree interval from there, and the calculator flags this rather than showing a misleading number. Enter your own angle, or use the default 30 degrees, to see the full set of results and how each is derived.

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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

Double Angle Results for 2x

sin(2x)
0.866025
= 2 sin x cos x

cos(2x)
0.500000
= cos²x − sin²x

tan(2x)
1.732051
= 2 tan x / (1 − tan²x)

Inputs used

Angle x30°

2x60°

sin x0.500000

cos x0.866025

tan x0.577350

All three forms of cos(2x)

cos²x − sin²x0.500000

2cos²x − 10.500000

1 − 2sin²x0.500000

Agreement checkAll three match

Summary: Enter an angle above.

What the double angle formulas are

The double angle formulas let you find the sine, cosine or tangent of twice an angle (2x) directly from the sine, cosine and tangent of the original angle (x), without having to look up or measure the doubled angle separately. They are a direct consequence of the angle sum formulas, sin(A + B) and cos(A + B), applied with A = B = x. The three identities are:

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

Why cos(2x) has three equivalent forms

The base identity cos(2x) = cos²x − sin²x can be rewritten using the Pythagorean identity sin²x + cos²x = 1. Replacing sin²x with (1 − cos²x) gives cos(2x) = 2cos²x − 1. Replacing cos²x with (1 − sin²x) gives cos(2x) = 1 − 2sin²x. All three forms always produce the same numeric value for any angle x. You pick whichever form is easiest to work with: the cosine-only form is useful for half angle derivations, and the sine-only form is useful when sine is the variable you already know.

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
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. Applying the formulas: sin(2x) = 2 × 0.5 × 0.866025 = 0.866025, which matches sin(60°). cos(2x) = 0.866025² − 0.5² = 0.75 − 0.25 = 0.5, which matches cos(60°). The other two cos(2x) forms give the same 0.5: 2(0.75) − 1 = 0.5, and 1 − 2(0.25) = 0.5. tan(2x) = 2(0.577350) / (1 − 0.333333) = 1.154701 / 0.666667 = 1.732051, which matches tan(60°). Enter 30 degrees above to see this calculator return the same figures.

Common uses

Simplifying trigonometric expressions and proving identities in NCEA and university calculus courses
Integrating expressions like sin²x or cos²x by rewriting them with the cos(2x) forms (power-reduction)
Physics problems involving wave interference, oscillations and AC electrical signals, where doubled-frequency terms appear naturally
Deriving the half angle formulas, which are the double angle formulas rearranged and solved for sin(x/2) and cos(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
Results are rounded to the number of decimal places you select, but the underlying calculation uses full floating point precision

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.
Sin Cos Tan Calculator: sine, cosine and tangent of any angle.

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.