Here I introduce you to the factor formulae. These are identities, given without proof are useful when adding or subtracting two sine angles or cosine angles and creating one term from the two. Hence, factor formulae.

The examples which follow are typical of the kind of questions you can get that uses the factor formulae.

Factor Formulae

  • \sin A + \sin B \equiv 2\sin \dfrac{{A + B}}{2}\cos \dfrac{{A - B}}{2} \\
    \sin A - \sin B \equiv 2\cos \dfrac{{A + B}}{2}\sin \dfrac{{A - B}}{2} \\
    \cos A + \cos B \equiv 2\cos \dfrac{{A + B}}{2}\cos \dfrac{{A - B}}{2} \\
    \cos A - \cos B \equiv - 2\sin \dfrac{{A + B}}{2}\sin \dfrac{{A - B}}{2}


  1. \dfrac{{\cos S + \cos T}}{{\sin S - \sin T}} \equiv \cot \dfrac{{S - T}}{2}


  1. \dfrac{{\sin S - \sin T}}{{\sin S + \sin T}} \equiv \cot \dfrac{{S + T}}{2} \tan \dfrac{{S - T}}{2}


  1. \cos 3x + \cos 5x + \cos 7x \equiv \cos 5x(2\cos 2x + 1)