Confusing

Let the given expression be $\mathfrak G$ . I'll be using $\small{\color{#D61F06}{I. ~\cos (A+B)=-\cos C}(\text{By taking }\cos \text{ of } A+B=\pi-C)}$ $\small{\color{#D61F06}{II.~\tan A+\tan B+\tan C=\tan A\tan B\tan C}(\text{By taking }\tan \text{ of } A+B+C=\pi)}$ $\small{\color{#D61F06}{III. \csc y=\dfrac1{\sin y}}}$ and $\small{\color{#D61F06}{IIII.~\cos (x+y)=\cos x\cos y-\sin x\sin y}}$ .

$\large\mathfrak G=-\displaystyle\sum_{\text{cyc}}\left(\dfrac{\cos (B+C)}{\sin B \sin C}\right)$ $\large =\displaystyle\sum_{\text{cyc}}\left(1-\cot B\cot C\right)$ $\large =3-\underbrace{\displaystyle\sum_{\text{cyc}}\left(\dfrac{\tan A}{\tan A\tan B\tan C}\right)}_{\large 1}$

$\huge \therefore \mathfrak G=\boxed{\color{#007fff}{2}}$

PS: Don't write in comments: just assume $A=B=C=\dfrac{\pi}3$ and you are done XD