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{{ printedBook.courseTrack.name }} {{ printedBook.name }} 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: sin2θ=2sinθcosθcos2θ=cos_{2}θ−sin_{2}θcos2θ=2cos_{2}θ−1cos2θ=1−2sin_{2}θtan2θ=1−tan_{2}θ2tanθ $

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, knowing that $cos60_{∘}=21 ,$ the exact value of $cos120_{∘}$ 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 $120_{∘}$ as $2⋅60_{∘}.$ $cos120_{∘} =cos2⋅60_{∘}=2cos_{2}60_{∘}−1=2⋅41 −1=-21 $ Below, a proof of the first two identities is shown. The rest of the identities can be proven by following a similar reasoning.

Start by writing the Angle Sum Identity for sine and cosine. $sin(x+y)=sinxcosy+cosxsinycos(x+y)=cosxcosy−sinxsiny $ Let $x=θ$ and $y=θ.$ With this, $x+y$ becomes $2θ.$ Then, the two formulas above can be rewritten in terms of $θ.$ $sin2θ=sinθcosθ+cosθsinθcos2θ=cosθcosθ−sinθsinθ $ 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 $sin2θ$ is obtained.

$sin2θ=sinθcosθ+sinθcosθ⇓sin2θ=2sinθcosθ $

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.

$cos2θ=cosθcosθ−sinθsinθ⇓cos2θ=cos_{2}θ−sin_{2}θ $