[จูกัดเหลียง]

Alg
1.substituting $a=\sqrt{\tan A},b=\sqrt{\tan B},...d=\sqrt{\tan D}$ the given equality becomes $\sin^2 A+\sin^2 B+...\sin ^2 D=3$ and set $x=\sin^2 A...w=\sin^2 D$
It's suffice to show $$\prod_{cyc} \Big(\dfrac{1}{x}-1\Big)\le \frac{1}{81}=\frac{1}{(x+y+z+w)^4}$$
or equivalent to $xyzw\ge(y+z+w-2x)(x+y+z-2w)(z+w+x-2y)(w+x+y-2z)$
we put $x=\alpha+\beta+\gamma ,y=\beta+\gamma+\eta ,...w=\eta+\alpha+\beta$
It's enough to show $$(\alpha+\beta+\gamma )(\beta+\gamma+\eta)(\gamma+\eta+\alpha)(\eta+\alpha+\beta)\ge 81\alpha\beta\gamma\eta$$
Which is true by AM-GM $\alpha+\beta+\gamma \ge 3\sqrt[3]{\alpha\beta\gamma}$