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{{ printedBook.courseTrack.name }} {{ printedBook.name }} To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.

$sin(x+y)cos(x+y)tan(x+y) =sinxcosy+cosxsiny=cosxcosy−sinxsiny=1−tanxtanytanx+tany $

There are also similar identities for the difference of two angles.

$sin(x−y)cos(x−y)tan(x−y) =sinxcosy−cosxsiny=cosxcosy+sinxsiny=1+tanxtanytanx−tany $

These identities are useful to find the exact value of the sine, cosine, or tangent at a given angle. For example, the exact value of $sin120_{∘}$ can be found by using the first formula. The first step is, in this case, to rewrite $120_{∘}$ as $90_{∘}+30_{∘}.$ $sin120_{∘} =sin(90_{∘}+30_{∘})=sin90_{∘}cos30_{∘}+cos90_{∘}sin30_{∘}=1⋅23 +0⋅21 =23 $ Below, a proof of the first identity is shown. The rest of the identities can be proven using similar reasoning.

Let $△AFD$ be a right triangle with hypotenuse $1$ and an acute angle with measure $x+y.$

By definition, the sine of an angle is the quotient between the length of the opposite side and the hypotenuse. $sin(x+y)=1DF ⇓DF=sin(x+y) $ Next, by drawing a ray, $∠A$ can be divided into two angles with measures $x$ and $y.$ Let $C$ be a point on this ray such that $△ACD$ and $△ABC$ are right triangles.

Consider $△ACD.$ By calculating the sine and cosine of $x,$ the legs of this triangle can be rewritten. $sinx=1DC ⇒DC=sinxcosx=1AC ⇒AC=cosx $ Consider $△ABC.$ Knowing that $AC=cosx,$ the sine of $y$ can be used to write $BC$ in terms of $x$ and $y.$ $sinysiny =ACBC =cosxBC ⇒BC=cosxsiny $ Let $G$ be the point of intersection between $FD$ and $AC.$ Notice that $∠AGF≅∠DGC$ by the Vertical Angles Theorem.

By the Third Angle Theorem, it is known that $∠GAF≅∠GDC.$ Therefore, $m∠GDC=y.$

Now, let $E$ be a point on $FD$ such that $EC∥AB.$ This implies that $EF=cosxsiny.$ In addition, $△CED$ is a right triangle and then $cosy$ can be calculated using $△CED.$

Finally, by the Segment Addition Postulate, $DF$ is equal to the sum of $DE$ and $EF.$ All these lengths have been rewritten in terms of the sine and cosine of $x$ and $y.$