$\displaystyle
\begin{gathered}
\sin (x \pm y) = \sin (x)\cos (x) \pm \sin (y)\cos (y) \hfill \\
\cos (x \pm y) = \cos (x)\cos (y) \mp \sin (x)\sin (y) \hfill \\
\hfill \\
{\text{Example:}} \hfill \\
{\text{ - find sin(15):}} \hfill \\
\sin (15) = \sin (45 - 30) \hfill \\
\end{gathered}
$
$\displaystyle
\begin{gathered}
\sin (15) = \sin (45)\cos (30) - \sin (30)\cos (45) \hfill \\
\sin (15) = \frac{{\sqrt 2 }}
{2} \cdot \frac{{\sqrt 3 }}
{2} - \frac{1}
{2} \cdot \frac{{\sqrt 2 }}
{2} \hfill \\
\boxed{\sin (15) = \frac{{\sqrt 2 }}
{2}\left( {\frac{{\sqrt 3 }}
{2} - \frac{1}
{2}} \right)} \hfill \\
\end{gathered}
$
$\displaystyle
\begin{gathered}
{\text{ - find cos(75)}} \hfill \\
{\text{cos(75)}} = \cos (45 + 30) \hfill \\
\cos (75) = \cos (45)\cos (30) - \sin (45)\cos (30) \hfill \\
etc... \hfill \\
\end{gathered}
$
but for this case especial:
$\displaystyle
\begin{gathered}
\sin (2x) = \sin (x + x) = 2\sin (x)\cos (x) \hfill \\
\cos (2x) = \cos (x + x) = \cos ^2 (x) - \sin ^2 (x) \hfill \\
\end{gathered}
$
that are Half angle formulas and I show you the born of this formulas, the Half angle formulas born too for this
$\displaystyle
\cos (x) = \cos \left( {\frac{x}
{2} + \frac{x}
{2}} \right) = \cos ^2 \left( {\frac{x}
{2}} \right) - \sin ^2 \left( {\frac{x}
{2}} \right) = \cos ^2 \left( {\frac{x}
{2}} \right) - \left( {1 - \cos ^2 \left( {\frac{x}
{2}} \right)} \right)
$
$\displaystyle
\begin{gathered}
\cos (x) = 2\cos ^2 \left( {\frac{x}
{2}} \right) - 1 \hfill \\
\cos \left( {\frac{x}
{2}} \right) = \sqrt {\frac{{\cos (x) + 1}}
{2}} \hfill \\
\hfill \\
but: \hfill \\
\end{gathered}
$
$\displaystyle
\begin{gathered}
\cos (x) = \cos ^2 \left( {\frac{x}
{2}} \right) - \sin ^2 \left( {\frac{x}
{2}} \right) = 1 - \sin ^2 \left( {\frac{x}
{2}} \right) - \sin ^2 \left( {\frac{x}
{2}} \right) \hfill \\
\cos (x) = 1 - 2\sin ^2 \left( {\frac{x}
{2}} \right) \hfill \\
\end{gathered}
$
$\displaystyle
\sin \left( {\frac{x}
{2}} \right) = \sqrt {\frac{{1 - \cos (x)}}
{2}}
$
now for Sum or Difference as a Product, is only an identity
$\displaystyle
\begin{gathered}
\sin (x) + \sin (y) = 2\sin \left( {\frac{{x + y}}
{2}} \right)\cos \left( {\frac{{x - y}}
{2}} \right) \hfill \\
\sin (x) - \sin (y) = 2\sin \left( {\frac{{x - y}}
{2}} \right)\cos \left( {\frac{{x + y}}
{2}} \right) \hfill \\
\end{gathered}
$
$\displaystyle
\begin{gathered}
\cos (x) + \cos (y) = 2\cos \left( {\frac{{x + y}}
{2}} \right)\cos \left( {\frac{{x - y}}
{2}} \right) \hfill \\
\cos (x) - \cos (y) = - 2\sin \left( {\frac{{x + y}}
{2}} \right)\sin \left( {\frac{{x - y}}
{2}} \right) \hfill \\
\end{gathered}
$
to tangent is easily deducible but if you want that i post the formulas, only ask for it