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The trigonometric angle addition identities state the following identities:

$\sin(x + y) = \sin (x) \cos (y) + \cos (x) \sin (y)$

$\cos(x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$

$\tan(x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)}$

$[asy] unitsize(216); real d = 1/cos(radians(35)); real d1 = d * cos(radians(55)); real d2 = d * sin(radians(55)); pair O = (0,0); pair A = (cos(radians(20)),0); pair B = (cos(radians(20)),sin(radians(20))); pair C = (cos(radians(20)),d2); pair D = (d1,d2); draw(O--A--B--O--D--B--O--D--C--B); dot(O); dot(B); dot(A,red); dot(C,green); dot(D,blue); label("O",O,SW); label("\alpha",shift(dir(10)/5)*O); label("\beta",shift(dir(37.5)/5)*O); label("A",A,SE,red); label("B",B,E); label("C",C,NE,green); label("D",D,dir(122.5),blue); label("\cos \alpha",O--A,S); label("\sin \alpha",A--B,E); label("1",O--B,dir(302.5)); label("\frac{\cos \alpha \sin \beta}{\cos \beta}",B--C,E); label("\frac{\sin \alpha \sin \beta}{\cos \beta}",C--D,N); label("\frac{\sin \beta}{\cos \beta}",B--D,dir(200)); label("\frac{1}{\cos \beta}",D--O,dir(325)); [/asy]$
$\fontsize{18}{27}\selectfont \sin (\alpha + \beta ) = \frac{\left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \sin \alpha + \frac{\cos \alpha \sin \beta}{\cos \beta} \right) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$
$\fontsize{18}{27}\selectfont \cos (\alpha + \beta ) = \frac{\left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right)}{\frac{1}{\cos \beta}} = \cos \beta \times \left( \cos \alpha - \frac{\sin \alpha \sin \beta}{\cos \beta} \right) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$
$\fontsize{18}{27}\selectfont \tan (\alpha + \beta ) = \frac{\sin (\alpha + \beta )}{\cos (\alpha + \beta )} = \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta - \sin \alpha \sin \beta} = \frac{\frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta}}{\frac{\cos \alpha \cos \beta - \sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\frac{\sin \alpha}{\cos \alpha} + \frac{\sin \beta}{\cos \beta}}{1 - \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}} = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$