Given $A+B+C=180^{\circ}$, find value of $\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$ 
Given $A+B+C=180^{\circ}$, find value of $$\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$$

I know about some basic conditional identities but don't know how to use them here.

Observe that 
$$\cos{A+B+C}=\cos{A}\cos{B}\cos{C}-\sin{A}\sin{B}\cos{C}-\sin{A}\cos{B}\sin{C}-\cos{A}\sin{B}\sin{C}\tag{1}$$
Let $k=\tan A\cdot\tan B+\tan B\cdot\tan C+\tan A\cdot \tan C-\sec A\cdot\sec B\cdot\sec C$
Divide $(1)$ by $\cos^2{A}\cos^2{B}\cos^2{C}$, to get the $-k=\cos{\pi}=-1 \Rightarrow k=1$
