由tan的和角公式：tan(x+y)=(tanx+tany)/(1-tanxtany)
所以 tanx+tany=tan(x+y)(1-tanxtany).
从而 tan7+tan8=tan15(1-tan7tan8),代入所求式得到
(1-tan7-tan8-tan7tan8)/(1+tan7+tan8-tan7tan8)
=(1-tan7tan8-tan15(1-tan7tan8))/(1-tan7tan8+tan15(1-tan7tan8))
=[(1-tan7tan8)(1-tan15)]/[(1-tan7tan8)(1+tan15)]
=(1-tan15)(1+tan15)
=(1-sin15/cos15)/(1+sin15/cos15)
=(cos15-sin15)/(cos15+sin15) (分子分母同时乘以cos15-sin15)
=(cos15-sin15)^2/[(cos15+sin15)(cos15-sin15)] (分母用平方差公式)
=(1-2sin15cos15)/[(cos15)^2-(sin15)^2] (分子分母同时用倍角公式)
=(1-sin30)/(cos30)
=根号3/3
即原式=根号3/3.