b→ and c→ are unit vectors. Then for any arbitrary vector a→,(((a→×b→)+(a→×c→))×(b→×c→)) ⋅(b→−c→) is always equal to

((a→×b→)+(a→×c→))×(b→×c→) =(a→×b→)×(b→×c→)+(a→×c→)×(b→×c→)=((a→×b→)⋅c→)b→−((a→×b→)⋅b→)c+((a→×c→)⋅c→)b→−((a→×c→)⋅b→)c→→=[a→b→c→](b→+c→)⇒ (a→×b→)+(a→×c→))×(b→×c→))⋅(b→−c→)=[a→b→c→](b→+c→)⋅(b→−c→)=[a→b→c→]|b→|2−|c|2=0