Để phương trình \[\frac{{{a^2}}}{{1 - {{\tan }^2}x}} = \frac{{{{\sin }^2}x + {a^2} - 2}}{{\cos 2x}}\] có nghiệm, tham số a phải thỏa mãn điều kiện:

\(\left\{ {\begin{array}{*{20}{c}}{1 - ta{n^2}x \ne 0}\\{cos2x \ne 0}\\{cosx \ne 0}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{\frac{{co{s^2}x - si{n^2}x}}{{co{s^2}x}}}\\{cos2x \ne 0}\\{cosx \ne 0}\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{cos2x \ne 0}\\{cosx \ne 0}\end{array}} \right.\)

\( \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{2x \ne \frac{\pi }{2} + k\pi }\\{x \ne \frac{\pi }{2} + k\pi }\end{array}} \right. \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{x \ne \frac{\pi }{4} + \frac{{k\pi }}{2}}\\{x \ne \frac{\pi }{2} + k\pi }\end{array}} \right.(k \in Z)\)

\[\frac{{{a^2}}}{{1 - ta{n^2}x}} = \frac{{si{n^2}x + {a^2} - 2}}{{cos2x}}\]

\( \Leftrightarrow \frac{{{a^2}}}{{\frac{{co{s^2}x - si{n^2}x}}{{co{s^2}x}}}} = \frac{{si{n^2}x + {a^2} - 2}}{{cos2x}}\)

\( \Leftrightarrow \frac{{{a^2}co{s^2}x}}{{cos2x}} = \frac{{si{n^2}x + {a^2} - 2}}{{cos2x}}\)

\[ \Leftrightarrow {a^2}co{s^2}x = si{n^2}x + {a^2} - 2\]

\[ \Leftrightarrow {a^2}co{s^2}x = 1 - co{s^2}x + {a^2} - 2\]

\[ \Leftrightarrow ({a^2} + 1)co{s^2}x = {a^2} - 1 \Leftrightarrow co{s^2}x = \frac{{{a^2} - 1}}{{{a^2} + 1}} < 1\]

Vì \[\cos x \ne 0 \Rightarrow 0 < {\cos ^2}x \le 1 \Leftrightarrow {\cos ^2}x >0 \Leftrightarrow {a^2} - 1 >0 \Rightarrow \left| a \right| >1\]