(sinx)(cosx)=1/2cosx
how do you solve this for the degree measures?

The
(Sinx)(Cosx)=1/2Cosx implies Sinx=1/2Cos²x, but we know Cos²x + sin²x =1
so Sinx=1/2Cos²x = 1/2(1-sin²x) this equation is equivalent to 
sin²x + 2sinx -1 = 0, let X = sinx, so x = arcsinX
X²+2X -1= 0 , and X²+2X -1=(X+1-sqrt2)(X+1+sqrt2)=0
so X= - 1- sqrt2 or  X= - 1+sqrt2

then x= arcsin(- 1+sqrt2)=24.46°