\[\begin{array}{l} \frac{1}{{\sin x}} + \frac{1}{{\cos x}} = \frac{{35}}{{12}} \Leftrightarrow 12\left( {\cos x + \sin x} \right) = 35\sin x\cos x \hfill \\ \cos x + \sin x = t \hfill \\ \sin x\cos x = \frac{{{t^2} - 1}}{2} \hfill \\ \left[ \begin{array}{l} \cos x + \sin x = - \frac{5}{7} \hfill \\ \cos x + \sin x = \frac{7}{5} \hfill \\ \end{array} \right. \hfill \\ \end{array}\]