Sine and cosine inequalities

Relevant wiki: Half Angle Tangent Substitution

$\begin{aligned} \frac {\sin x}{\cos x -1} & \ge 1 & \small \color{#3D99F6} \text{Let }t = \tan \frac x2 \\ \frac {\frac {2t}{1+t^2}}{\frac {1-t^2}{1+t^2}-1} & \ge 1 \\ - \frac 1t & \ge 1 \\ \implies -1 & \le t < 0 & ...(1)\end{aligned}$

Similarly,

$\begin{aligned} \frac {3\cos x-1}{\sin x} & \ge 1 \\ \frac {3-3t^2-1-t^2}{2t} & \ge 1 \\ \frac {2-4t^2}{2t} & \ge 1 \\ 2t^2+t - 1 & \ge 0 \\ (t+1)(2t-1) & \ge 0 \\ \implies t \le -1 \cup t & \ge \frac 12 & ...(2) \end{aligned}$

Comparing $(1)$ and $(2)$ , we note that there is only one solution $t= \tan \dfrac x2 = -1 \implies x = \boxed{270}^\circ$ ,