If $$\cos x = \frac{-\sqrt3}{2}  and  \pi < x < \frac{3\pi}{2}$$, then the value of $$4 \cot^2 x - 3 \cosec^2 x$$ is:

As Given in Question :

$$\cos x =  \frac {-\sqrt3}{2}$$

$$\Rightarrow \cos x = -\cos 30^{\circ}$$

$$\Rightarrow \cos x = -\cos 30^{\circ} = \cos (180 + 30)^{\circ}$$     [ $$\because -\cos \theta = \cos (180 + \theta ) ]$$

$$\therefore \cos x = \cos 210^{\circ}$$

$$\Rightarrow x=210^{\circ}$$

Now  $$4 \cot^2 x - 3 \cosec^2 x$$

$$\Rightarrow 4 \cot^2 210^{\circ} - 3 \cosec^2 210^{\circ}$$

$$\Rightarrow 4 (\sqrt3)^2 - 3 (-2)^2 $$  $$ [ \because \cot (180 + \theta) = \cot \theta , \cosec (180+\theta) = -\cosec \theta] $$

$$\Rightarrow 4\times3 - 3\times4$$

$$\Rightarrow 12 - 12 = 0$$