$\begin{array}{l} \sin \left( {a + b} \right)\sin \left( {a - b} \right)\\ = \left( {\sin a\cos b + \sin b\cos a} \right)\left( {\sin a\cos b - \sin b\cos a} \right)\\ = {\left( {\sin a\cos b} \right)^2} - {\left( {\sin b\cos a} \right)^2}\\ = {\sin ^2}a\left( {1 - {{\sin }^2}b} \right) - \left( {1 - {{\sin }^2}a} \right){\sin ^2}b\\ = {\sin ^2}a - {\sin ^2}b \end{array}$
f)
$\begin{array}{l} \cos \left( {\dfrac{\pi }{6} + a} \right) + \cos \left( {\dfrac{\pi }{6} - a} \right)\\ = 2\cos \left( {\dfrac{{\dfrac{\pi }{6} + a + \dfrac{\pi }{6} - a}}{2}} \right)\cos \left( {\dfrac{{\dfrac{\pi }{6} + a - \dfrac{\pi }{6} + a}}{2}} \right)\\ = 2\cos \dfrac{\pi }{6}\cos a = 2.\dfrac{{\sqrt 3 }}{2}\cos a = \sqrt 3 \cos a \end{array}$
