It may be a little roundabout, but here goes:
Using the
product sum identity,
sin(x)  sin(y) = 2*cos((x+y)/2)*sin((xy)/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 20091225 07:24:35 