Double Angle Identities
A focused guide to sine, cosine, and tangent double-angle identities and how to use them in simplification.
Quick Answer
- Short definition
- Double-angle identities are exact equations that rewrite functions of 2θ using θ only.
- Formula
- cos(2θ) = 1 − 2 sin² θ
Introduction
Double Angle Calculator evaluates all three identity outputs at once for any θ you enter. This article focuses on using the identities as transformation tools, not just for finding numbers, but for rewriting expressions in proofs, equations, and calculus.
Identities are equations that hold for every valid angle. Double-angle identities are no exception. They let you move between equivalent forms without changing the underlying value.
If you need the raw formula list first, open Double Angle Formula. To see how double-angle identities differ from the half-angle family, read Double Angle vs Half Angle Formulas before working the simplification steps below.
Main Content
What is it?
Identities are transformation rules. They let you move from sin(2θ) to 2 sin θ cos θ, or from cos(2θ) to cos² θ − sin² θ, without changing the value. In a proof, that move may expose a factor you can cancel. In calculus, it may convert a product into a single trig function you can integrate.
Trigonometric simplification often starts by spotting 2θ in the argument or a product sin θ cos θ on the other side of the equation. Either pattern suggests a double-angle step. Training your eye to see those patterns saves time on exams.
The three cosine forms are the same identity in different costumes. cos(2θ) = cos² θ − sin² θ highlights symmetry. cos(2θ) = 2 cos² θ − 1 helps when cos θ is known. cos(2θ) = 1 − 2 sin² θ appears when sin² θ is already in the expression.
Tangent’s identity comes with domain restrictions. Whenever you rewrite tan(2θ), note angles where 1 − tan² θ = 0 or where cos(2θ) = 0 in an equivalent fraction form.
Formula
- sin(2θ) = 2 sin θ cos θ
- cos(2θ) = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ
- tan(2θ) = 2 tan θ / (1 − tan² θ)
Reverse uses matter too. From cos(2θ) = 1 − 2 sin² θ you can solve for sin² θ:
sin² θ = (1 − cos(2θ)) / 2
That rearrangement is the bridge to half-angle formulas, which we compare directly in Double Angle vs Half Angle Formulas.
Step-by-step guide
- Spot 2θ inside the function argument. Or spot sin θ cos θ, which converts to sin(2θ)/2.
- Match the function to its identity. Do not apply a cosine form to a sine target.
- Replace with the θ-only form. Write the identity explicitly before substituting.
- Simplify the algebra. Factor, cancel, and apply the Pythagorean identity if squared terms remain.
- Check numerically when possible. Pick a sample θ and compare with the calculator or Double Angle Examples.
Example
Example 1: product: Simplify sin(40°) cos(40°).
2 sin(40°) cos(40°) = sin(80°), so sin(40°) cos(40°) = sin(80°)/2.
Example 2: squared sine: Rewrite 1 − 2 sin²(3x).
1 − 2 sin²(3x) = cos(6x). The identity cos(2θ) = 1 − 2 sin² θ with θ = 3x gives the result in one step.
Example 3: equation: Solve sin(2x) = cos(x) on [0, 2π).
Substitute sin(2x) = 2 sin x cos x to get 2 sin x cos x = cos x, then factor cos x. Solutions follow from cos x = 0 or sin x = 1/2.
FAQ
Are identities the same as formulas?
In practice yes: double-angle formulas are the standard double-angle identities used in textbooks.
When should I rewrite toward 2θ vs away from 2θ?
Rewrite toward 2θ when a product or sum pattern suggests a single frequency. Rewrite away from 2θ when you need sin θ or cos θ alone for substitution or integration.
Conclusion
Master the three function families, then connect geometric intuition on the unit circle to algebraic moves in proofs.
Continue with Double Angle in Trigonometry for graph and triangle context, or Double Angle Identities on the Unit Circle for a coordinate-based view of the same equations.