Properties Of Triangles

The governing principle is the angle-sum constraint that the three interior angles of any triangle satisfy $A+B+C=\pi$, which is precisely the conditional restriction that produces a remarkable tangent identity not valid for arbitrary angles. Starting from $A+B=\pi -C$ and taking the tangent of both sides, the supplementary-angle rule gives $\tan (A+B)=\tan (\pi -C)=-\tan C$. Now expand the left side with the tangent addition formula: $\frac{\tan A+\tan B}{1-\tan A\tan B}=-\tan C$. Cross-multiplying clears the denominator, yielding $\tan A+\tan B=-\tan C(1-\tan A\tan B)$, which expands to $-\tan C+\tan A\tan B\tan C$. Moving the $\tan C$ term to the left side gives the elegant result $\tan A+\tan B+\tan C=\tan A\tan B\tan C$, valid whenever no angle is a right angle so that each tangent is finite. Option $\cot A+\cot B+\cot C$ corresponds to a different cotangent relation and is unrelated to this sum. Option with the pairwise products equalling the sum holds only under other constraints, not the triangle condition. Option $1$ is a baseless numerical guess. This is the classic conditional triangle tangent identity of JEE Advanced. As a final check, taking the equilateral case $A=B=C=60^{\circ }$ gives left side $3\sqrt{3}$ and right side $(\sqrt{3}{)}^3=3\sqrt{3}$, which agree, confirming the identity.