Explanation

(a)
$\frac{\overrightarrow{\mathbf{a}} \bullet \left( \ \overrightarrow{\mathbf{b}} \times \ \overrightarrow{\mathbf{c}} \right)}{\overrightarrow{\mathbf{b}} \bullet \left( \ \overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}} \right)} + \frac{\overrightarrow{\mathbf{b}} \bullet \left( \ \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} \right)}{\overrightarrow{\mathbf{a}} \bullet \left( \overrightarrow{\mathbf{b}} \times \ \overrightarrow{\mathbf{c}} \right)}$
$= \frac{\ \lbrack\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\rbrack}{\lbrack\overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\ \overrightarrow{\mathbf{a}}\rbrack} + \frac{\lbrack\overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\rbrack}{\lbrack\overrightarrow{\mathbf{a}}\ \overrightarrow{\mathbf{b}}\ \overrightarrow{\mathbf{c}}\rbrack} = 1 + 0 = 1$