分类：教育/科学-理工学科-数学
问题：证明题若n∈N+,证明:[tan(π/12)]^2+[cot(π 
若n∈N+,证明:[tan(π/12)]^2+[cot(π/12)]^2为一正偶数.
回答：证明:
设an=[tan(π/12)]^n+[cot(π/12)]^n,
易知,tan(π/12)+cot(π/12)=4,tan(π/12)cot(π/12)=1.
故tan(π/12)、cot(π/12)是
x^2-4x+1=0的两根.
{(tan(π/12))^2-4tan(π/12)+1=0 …… (1)
{(cot(π/12))^2-4cot(π/12)+1=0 …… (2)
由(1)×(tan(π/12))^n+(2)×(cot(π/12))^n,
得a<n+2>=4a<n+1>-a<n>,
由于
a<1>=4,
a<2>=[tan(π/12)+cot(π/12)-2=14.
∴由数学归纳法知a<n>是偶数,命题得证.