Rút gọn biểu thức A = (sin^2 α + cos α)/(1 − cos^2 α + sin α).

a) \(A = \frac{{\sin 2\alpha + \cos \alpha }}{{1 - \cos 2\alpha + \sin \alpha }} = \frac{{2\sin \alpha \cos \alpha + \cos \alpha }}{{2{{\sin }^2}\alpha + \sin \alpha }} = \frac{{\cos \alpha \left( {2\sin \alpha + 1} \right)}}{{\sin \alpha \left( {2\sin \alpha + 1} \right)}} = \frac{{\cos \alpha }}{{\sin \alpha }} = \cot \alpha \).

b) \(\sin \left( {3x + \frac{\pi }{3}} \right) = 0 \Leftrightarrow 3x + \frac{\pi }{3} = k\pi \Leftrightarrow 3x = - \frac{\pi }{3} + k\pi \Leftrightarrow x = - \frac{\pi }{9} + \frac{{k\pi }}{3}\quad \left( {k \in \mathbb{Z}} \right)\).