Analytically prove sin(54)-sin(18) = 1/2

It may be a little roundabout, but here goes:

Using the product sum identity,
sin(x) - sin(y) = 2*cos((x+y)/2)*sin((x-y)/2),
we have for x=54 and y=18:
sin(54) - sin(18) = 2*cos(36)*sin(18)

Using the double angle formula,
sin(2t) = 2*sin(t)*cos(t),
we have for t=18:
sin(36) = 2*sin(18)cos(18), therefore
sin(18) = sin(36)/(2*cos(18))

Substituting back into the earlier equation and simplifying:
sin(54) - sin(18) = cos(36)*sin(36)/cos(18)

Using the complement of the trig function,
cos(t) = sin(90 - t),
we have for t=18:
sin(54) - sin(18) = cos(36)*sin(36)/sin(72)

Reapplying the double angle formula and substituting, with t=36, we have:
sin(54) - sin(18) = cos(36)*sin(36)/[2*sin(36)cos(36)]

Simplifying:    
sin(54) - sin(18) = 1/2

Posted by Dej Mar on 2009-12-25 07:24:35