Angle Addition Formulas (Trigonometry)

$[asy] real a,b,c; a=0.4924; b=0.5467; c=a+b; pair A,B,C,D,E,F; A=(0,0); D=(cos(c),sin(c)); B=(cos(a)*cos(b),0); C=(cos(a)*cos(b),sin(a)*cos(b)); draw(A--B--C--D--cycle); draw(A--C); E=foot(D,A,B); draw(D--E); F=foot(C,D,E); draw(C--F); draw(anglemark(B,A,C,4)); draw(anglemark(C,A,D,5.5)); label("$\alpha$", A ,7*dir(a*30)); label("$\beta$", A ,9*dir(b*45+ a*30)); [/asy]$