5. (a) Using the formulae
$$\begin{gathered}
\sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B
\cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B
\end{gathered}$$
show that
- \(\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B\),
- \(\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B\).
(b) Use the above results to show that
$$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$
Using the result of part (b) and the exact values of \(\sin 60 ^ { \circ }\) and \(\cos 60 ^ { \circ }\),
(c) find an exact value for \(\cot 75 ^ { \circ }\) in its simplest form.