High School Trigonometry/Product-and-Sum, Sum-and-Product, and Linear Combinations of Identities

Product to Sum Formulas $$sin(u)sin(v)=\frac{1}{2}[cos(u-v)-cos(u+v)]$$ $$cos(u)cos(v)=\frac{1}{2}[cos(u-v)+cos(u+v)]$$ $$sin(u)cos(v)=\frac{1}{2}[sin(u+v)+sin(u-v)]$$ $$cos(u)sin(v)=\frac{1}{2}[sin(u+v)-sin(u-v)]$$ Sum to Product Formula $$sin(u)+sin(v)=2sin(\frac {u+v}{2})cos(\frac{u-v}{2})$$ $$sin(u)-sin(v)=2cos(\frac {u+v}{2})sin(\frac{u-v}{2})$$ $$cos(u)+cos(v)=2cos(\frac {u+v}{2})cos(\frac{u-v}{2})$$ $$cos(u)-cos(v)=-2sin(\frac{u+v}{2})sin(\frac{u-v}{2})$$