$\begin{array}{l} A = \left( {1 + 2\cos 2a + 2\cos 4a + 2\cos 6a} \right).\sin a\\ A = \sin a + 2\cos 2a.\sin a + 2\cos 4a.\sin a + 2\cos 6a.\sin a\\ A = \sin a + \dfrac{1}{2}.2\left( {\sin 3a - \sin a} \right) + \dfrac{1}{2}.2\left( {\sin 5a - \sin 3a} \right) + \dfrac{1}{2}.2\left( {\sin 7a - \sin 5a} \right)\\ A = \sin a + \sin 3a - \sin a + \sin 5a - \sin 3a + \sin 7a - \sin 5a\\ A = \sin 7a \end{array}$
