((sin(x)+sin(3x))/(2sin(2x)))=cos(x)

Hint: to get started use the identity sin(a+b)=sin(a)cos(b)-sin(b)cos(a), where a = x and b = 2x. Then the idenity cos(2x) = 2cos^2x-1 and sin(2x)=2sin(x)cos(x). The rest is just manipulation.