Double Angle Identities

Doubling an angle does not double its sine or cosine; it creates specific products and squares.

Sine double-angle identity

Set $A=B=\theta$ in the sine sum identity:

Thus

Cosine double-angle identities

The cosine sum identity gives

Using $\sin^2\theta+\cos^2\theta=1$ produces two equivalent forms:

Choose the form containing the known ratio or the expression you need to simplify.

Tangent double-angle identity

From the tangent sum identity,

where the expression is defined. A zero denominator corresponds to a doubled angle whose tangent is undefined.

Worked example from a triangle ratio

The signs match because $2\theta$ is acute for the triangle represented here.

Quadrant reasoning

Knowing the quadrant of $\theta$ does not automatically give the quadrant of $2\theta$. Determine a range for $2\theta$ before assigning signs.

For example, if $100^\circ<\theta<120^\circ$, then $200^\circ<2\theta<240^\circ$, placing the doubled angle in Quadrant III.

Choosing a cosine form

If only $\sin\theta$ is known, $1-2\sin^2\theta$ avoids finding cosine. If only cosine is known, use $2\cos^2\theta-1$. If both are known, $\cos^2\theta-\sin^2\theta$ may be direct.

All three are identical, so differing results reveal an arithmetic or sign error.

Power-reduction identities

Rearrange cosine double-angle forms:

These reduce squared trig functions and are useful in integration, identity verification, and modelling periodic power.

Solving equations

Double-angle identities can convert an equation to one trig function. For example,

may turn an equation involving both $\cos2x$ and $\sin x$ into a quadratic in $\sin x$. After solving, recover every angle in the stated interval.

Period awareness

The function $\sin2x$ completes twice as many cycles as $\sin x$ over the same interval. When solving, first solve for the compound angle $2x$, include all of its solutions, and then divide by $2$.

Dividing too early or listing only one cycle loses solutions.

Common mistakes

Writing $\sin2\theta=2\sin\theta$. The cosine factor is required.

Writing $\cos2\theta=2\cos\theta$. Use one of the three squared forms.

Assuming $2\theta$ stays in the same quadrant. Double the angle range.

Using a cosine form that introduces an unknown unnecessarily. Choose strategically.

Forgetting domain restrictions in tangent. Its denominator and the original tangent must be defined.

Quick self-check

• Which double-angle identity matches the known information?
• Is the doubled angle's quadrant determined separately?
• Are squared fractions and signs handled accurately?
• If solving, have all cycles of the compound angle been included?
• Can another equivalent cosine form verify the result?

Related topics

• Sum and Difference Identities
• Trigonometric Identities
• Solving Trigonometric Equations
• Exact Trigonometric Values