化简2sin^2(x)sin^2(φ)+2cos^2(x)cos^2(φ)-cos2(x)cos2(φ)

题目

化简2sin^2(x)sin^2(φ)+2cos^2(x)cos^2(φ)-cos2(x)cos2(φ)

答案

最后的乘积项用二倍角公式展开
原式＝2sin^2(x)sin^2(φ)+2cos^2(x)cos^2(φ)-[cos^2(x)-sin^2(x)][cos^2(φ)-sin^2(φ)]
=2sin^2(x)sin^2(φ)+2cos^2(x)cos^2(φ)-[cos^2(x)cos^2(φ)-cos^2(x)sin^2(φ)-sin^2(x)cos^2(φ)+
sin^2(x)sin^2(φ)]
合并同类项 =sin^2(x)sin^2(φ)+cos^2(x)cos^2(φ)+cos^2(x)sin^2(φ)+sin^2(x)cos^2(φ)
　　＝sin^2(x)[sin^2(φ)+cos^2(φ)]+cos^2(x)[cos^2(φ)+sin^2(φ)]
　　＝sin^2(x)+cos^2(x)
　　＝1