Given,
tan9o−tan27o−tan63o+tan81o
=(cot81∘+tan81o)−(tan27∘+cot27o)
=(tan9∘+cot9∘)−(tan27∘+cot27∘)
=(cos9∘sin9∘+sin9∘cos9∘)−(cos27∘sin27∘+sin27∘cos27∘)
=sin18o2−sin54o2
=(5−12×4−5+12×4)
=8(5−11−5+11)
=8((5−1)(5+1)5+1−(5−1))
=8(5−12)=4