$
1)\,\,3 - 4{\cos ^2}x = {\mathop{\rm s}\nolimits} {\rm{inx}}(2\sin x + 1)\\
\Leftrightarrow 3 - 4\left( {1 - {{\sin }^2}x} \right) = {\mathop{\rm s}\nolimits}
{\rm{inx}}(2\sin x + 1)\\
\Leftrightarrow 4{\sin ^2}x - 1 = {\mathop{\rm s}\nolimits} {\rm{inx}}(2\sin x + 1)\\
\Leftrightarrow (2\sin x + 1)({\mathop{\rm s}\nolimits} {\rm{inx}} - 1) = 0\\$
$\Leftrightarrow\left[\begin{array}{I}\sin
x=\frac{-1}{2}\\\sin x=1\end{array}\right.$
$
\Rightarrow \left[ \begin{array}{l}
x = - \frac{\pi }{6} + 2k\pi \\
x = \frac{{7\pi }}{{6}} + 2k\pi \\
x = \frac{\pi }{2} + 2k\pi
\end{array} \right.
$
$\begin{array}{l}
2)\,\,a + c = 2b \Leftrightarrow 2R\sin A + 2R\sin B = 4R\sin B \Leftrightarrow \sin A + \sin C
= 2\sin B\\
\Leftrightarrow 2\sin \frac{{A + C}}{2}c{\rm{os}}\frac{{A - C}}{2} = 4\sin
\frac{B}{2}c{\rm{os}}\frac{B}{2}\\
\Leftrightarrow c{\rm{os}}\frac{B}{2}c{\rm{os}}\frac{{A - C}}{2} = 2\cos \frac{{A +
C}}{2}c{\rm{os}}\frac{B}{2}\\
\Leftrightarrow c{\rm{os}}\frac{{A - C}}{2} = 2\cos \frac{{A + C}}{2}\\
\Leftrightarrow c{\rm{os}}\frac{A}{2}c{\rm{os}}\frac{C}{2} + \sin \frac{A}{2}\sin
\frac{C}{2} = 2\cos \frac{A}{2}c{\rm{os}}\frac{C}{2} - 2\sin \frac{A}{2}\sin \frac{C}{2}\\
\Leftrightarrow 3\sin \frac{A}{2}\sin \frac{C}{2} =
c{\rm{os}}\frac{A}{2}c{\rm{os}}\frac{C}{2}\\
\Leftrightarrow \cot \frac{A}{2}\cot \frac{C}{2} = 3
\end{array}$