doublesin[Sin[function_]] := 2 Sin[#/2] Cos[#/2] &@ function
doublecos[Cos[function_]] := 1 - 2 Sin[#/2]^2 &@ function

In[278]:= doublesin[Sin[2 a]]

Out[278]= 2 Cos[a] Sin[a]

In[279]:= Sin[2 b] // doublesin

Out[279]= 2 Cos[b] Sin[b]

In[288]:= doublecos[Cos[4 c]]

Out[288]= 1 - 2 Sin[2 c]^2

In[289]:= Cos[4 c] Sin[4 a] // doublesin // doublecos

Out[289]= doublecos[doublesin[Cos[4 c] Sin[4 a]]]

In[291]:= Cos[4 c] Sin[4 a] // doublecos

Out[291]= doublecos[Cos[4 c] Sin[4 a]]