$\begin{array}{l} a)\cos 2x - {\sin ^2}5x = 1\\ \Leftrightarrow \cos 2x = 1 + {\sin ^2}5x\\ + \cos 2x \le 1,{\sin ^2}5x \ge 0\\ \Rightarrow \cos 2x \le 1,1 + {\sin ^2}5x \ge 1\\ \Rightarrow \left\{ \begin{array}{l} \cos 2x = 1\\ {\sin ^2}5x = 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} 2x = k2\pi \\ 5x = k\pi \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = k\pi \\ x = \dfrac{{k\pi }}{5} \end{array} \right.\\ \Rightarrow x = k\pi \left( {k \in \mathbb{Z}} \right)\\ b){\tan ^2}x + {\cot ^2}x = {\tan ^2}x + \dfrac{1}{{{{\tan }^2}x}} \ge 2\\ + 2{\sin ^3}\left( {x + \dfrac{\pi }{4}} \right) \le 2\\ PT:2{\sin ^3}\left( {x + \dfrac{\pi }{4}} \right) = {\tan ^2}x + {\cot ^2}x\\ \Rightarrow \left\{ \begin{array}{l} {\tan ^2}x = 1\\ {\sin ^3}\left( {x + \dfrac{\pi }{4}} \right) = 1 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} \tan x = \pm 1\\ \sin \left( {x + \dfrac{\pi }{4}} \right) = 1 \end{array} \right.\\ \Leftrightarrow \left\{ \begin{array}{l} x = \pm \dfrac{\pi }{4} + k\pi \\ x + \dfrac{\pi }{4} = \dfrac{\pi }{2} + k2\pi \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = \pm \dfrac{\pi }{4} + k\pi \\ x = \dfrac{\pi }{4} + k2\pi \end{array} \right.\\ \Rightarrow x = \dfrac{\pi }{4} + k2\pi \left( {k \in \mathbb{Z}} \right)\\ c)\sqrt 3 \sin x + \cos x = 3 + \sin 3x\\ \Leftrightarrow 2\sin \left( {x + \dfrac{\pi }{6}} \right) = 3 + \sin 3x\\ + 2\sin \left( {x + \dfrac{\pi }{6}} \right) \le 2,\sin 3x \ge - 1\\ \Rightarrow 2\sin \left( {x + \dfrac{\pi }{6}} \right) \le 2,3 + \sin 3x \ge 2\\ \Rightarrow \left\{ \begin{array}{l} \sin \left( {x + \dfrac{\pi }{6}} \right) = 1\\ \sin 3x = - 1 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x + \dfrac{\pi }{6} = \dfrac{\pi }{2} + k2\pi \\ 3x = - \dfrac{\pi }{2} + k2\pi \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = \dfrac{\pi }{3} + k2\pi \\ x = - \dfrac{\pi }{6} + \dfrac{{k2\pi }}{3} \end{array} \right.\\ \Rightarrow x \in \emptyset \\ d)4{\cos ^2}x + 3{\tan ^2}x - 4\sqrt 3 \cos x + 2\sqrt 3 \tan x + 4 = 0\\ \Leftrightarrow {\left( {2\cos x - \sqrt 3 } \right)^2} + 3{\tan ^2}x + 2\sqrt 3 \tan x + 1 = 0\\ \Leftrightarrow {\left( {2\cos x - \sqrt 3 } \right)^2} + {\left( {\sqrt 3 \tan x + 1} \right)^2} = 0\\ \Leftrightarrow \left\{ \begin{array}{l} \cos x = \dfrac{{\sqrt 3 }}{2}\\ \tan x = - \dfrac{{\sqrt 3 }}{3} \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = \pm \dfrac{\pi }{6} + k2\pi \\ x = - \dfrac{\pi }{6} + k\pi \end{array} \right. \Rightarrow x = - \dfrac{\pi }{6} + k\pi \left( {k \in \mathbb{Z}} \right) \end{array}$
