The value of tan9° − tan27° − tan63° + tan81° is:

Since 81° = 90° − 9° and 63° = 90° − 27°, the expression regroups as (tan9° + cot9°) − (tan27° + cot27°). Now tanx + cotx = 1/(sinx·cosx) = 2/sin2x, so the value is 2/sin18° − 2/sin54°. With sin18° = (√5 − 1)/4 and sin54° = cos36° = (√5 + 1)/4 this is 8/(√5 − 1) − 8/(√5 + 1) = $8\times 2/4=4$. Arithmetic check: 0.15838 − 0.50953 − 1.96261 + 6.31375 = 4.000. Treating the four terms as two cancelling complementary pairs gives the trap 0.