Rule

Double-Angle Identities

The double-angle identities are special cases of the angle sum identities. To evaluate trigonometric functions of the double of an angle, the following identities can be applied.

$Sine : Cosine : Tangent : sin 2 θ = 2 sin θ cos θ cos 2 θ = cos^{2} θ - sin^{2} θ cos 2 θ = 2 cos^{2} θ - 1 cos 2 θ = 1 - 2 sin^{2} θ tan 2 θ = \frac{2 tan θ}{1 - tan ^{2} θ}$

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that $cos 6 0^{\circ} = \frac{1}{2},$ the exact value of $cos 12 0^{\circ}$ can be found by using the second formula for the cosine of the double of an angle. In this case, the first step is to rewrite $12 0^{\circ}$ as $2 \cdot 6 0^{\circ} .$ $cos 12 0^{\circ} = cos 2 \cdot 6 0^{\circ} = 2 cos^{2} 6 0^{\circ} - 1 = 2 \cdot \frac{1}{4} - 1 = - \frac{1}{2}$ Below, a proof of the first two identities is shown. The rest of the identities can be proven by following a similar reasoning.

Proof

Start by writing the Angle Sum Identity for sine and cosine. $sin (x + y) = sin x cos y + cos x sin y cos (x + y) = cos x cos y - sin x sin y$ Let $x = θ$ and $y = θ .$ With this, $x + y$ becomes $2 θ .$ Then, the two formulas above can be rewritten in terms of $θ .$ $sin 2 θ = sin θ cos θ + cos θ sin θ cos 2 θ = cos θ cos θ - sin θ sin θ$ The Commutative Property of Multiplication can be applied to the second term of the right-hand side of the first equation. Then, by adding the terms on the right-hand side of this equation, the formula for $sin 2 θ$ is obtained.

$sin 2 θ = sin θ cos θ + sin θ cos θ ⇓ sin 2 θ = 2 sin θ cos θ$

Finally, the Product of Powers Property can be used to rewrite the right-hand side of the second equation. By doing this, the first identity for the cosine of the double of an angle is obtained.

$cos 2 θ = cos θ cos θ - sin θ sin θ ⇓ cos 2 θ = cos^{2} θ - sin^{2} θ$