Angle addition identities

The trigonometric angle addition identities state the following identities:

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Proofs

unitsize(216);
pair O = (0,0);
pair A = (cos(radians(20)),0);
pair B = (cos(radians(20)),sin(radians(20)));
pair C = (cos(radians(20)),sin(radians(55)));
pair D = ((cos(radians(55))*sin(radians(35))),(sin(radians(55))*sin(radians(35))));
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("$\frac{\sin \alpha \sin \beta}{\cos \beta}$",B--D,dir();
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See Also

• Trigonometric identities