compound angle, i think that's what it's called, where you expand the double angle inside sin and cos
=(sin45cosx-cos45sinx)(cos45cosx+sin45sinx)
=(1/sqroot(2)cosx-1/sqroot(2)sinx)(1/sqroot(2)cosx+1/sqroot(2)sinx)
simplify it
you get
1/sqroot(2)(cosx-sinx)(cosx+sinx)
and difference between...