$\begin{array}{l}
2\sin 2x + 1 = 0\\
\Leftrightarrow \sin 2x = - \dfrac{1}{2}\\
\Leftrightarrow \left[ \begin{array}{l}
2x = - {30^o} + k{360^o}\\
2x = {180^o} - \left( { - {{30}^o}} \right) + k2\pi
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = - {15^o} + k{180^o}\\
x = {105^o} + k{180^o}
\end{array} \right.\\
k = 1 \Rightarrow x = {165^o},x = {390^o} \notin \left( {{0^o};{{90}^o}} \right)\\
k = 0 \Rightarrow x = - {15^o} \notin \left( {{0^o};{{90}^o}} \right),x = {105^o} \notin \left( {{0^o};{{90}^o}} \right)\\
k = - 1 \Rightarrow x = - {195^o} \notin \left( {{0^o};{{90}^o}} \right),x = - {75^o} \notin \left( {{0^o};{{90}^o}} \right)\\
\Rightarrow x \in \emptyset \\
b){\cos ^3}x - 2{\cos ^2}x = 0\\
\Leftrightarrow {\cos ^2}x\left( {\cos x - 2} \right) = 0\\
\Leftrightarrow \cos x = 0\\
\Leftrightarrow x = {90^o} + k{180^o}\left( {k \in Z} \right)\\
x \in \left[ {{0^o};{{720}^o}} \right] \Rightarrow {0^o} \le {90^o} + k{180^o} \le {720^o}\\
\Leftrightarrow - \dfrac{1}{2} \le k \le \dfrac{7}{2} \Rightarrow k \in \left[ {0;3} \right]\\
\Rightarrow x = {90^o},x = {270^o},x = {450^o},x = {630^o}
\end{array}$
