Application of AM-GM Inequality in Trigonometry

In triangle \(ABC\), \(a\), \(b\) and \(c\) are the opposite sides of the interior angles \(A\), \(B\) and \(C\), respectively. It is known that \(a = \sqrt{3} \sin A\).

1. Show that \(\frac{abc}{2c^2 + 2ab \cos C} \leq \frac{\sqrt{3} \sin C}{4 - 2 \cos C}.\)

You may find the famous Inequality of Arithmetic and Geometric Means (AM-GM Inequality)
useful: For any non-negative real numbers \(x_1\) and \(x_2\), it holds that \[AM = \frac{x_1 + x_2}{2} \geq \sqrt{x_1 x_2} = GM,\]where \(AM\) and \(GM\) denotes the arithmetic mean and geometric mean of \(x_1\) and \(x_2\), respectively.

2. Hence, or otherwise, show that the maximum value of \(\frac{abc}{2c^2 + 2ab \cos C}\) is \(\frac{1}{2}\) and this maximum value is achieved when \(\Delta ABC\) is an equilateral triangle.