摘 要: | 题 1 若α、β、γ均为锐角 ,且满足cos2 α+cos2 β +cos2 γ=1.求证 :ctg2 α +ctg2 β+ctg2 γ≥ 32 .证明 如图 1,设以a、b、c为三度的长方体ABCD A1 B1 C1 D1 的对角线AC1 与三条棱AD、AB、AA1 所成角分别为α、β、γ ,则 ctgα=ADDC1=ab2 +c2 ,ctgβ=ABBC1 =ba2 +c2 , ctgγ=AA1 A1 C1=ca2 +b2 ,∴ ctg2 α +ctg2 β+ctg2 γ =a2b2 +c2 +b2a2 +c2 +c2a2 +b2 =a2 +b2 +c2b2 +c2 +a2 +b2 +c2a2 +c2 +a2 +b2 +c2a2 +b2 -3 =(a2 +b2 +c2 ) ( 1b2 +c2 +1a2 +c2 +1a2 +b2 ) -3 =12 [(b2 +c2 ) +(a2 +c2 ) +(a2 +b2 ) ]&;#183;( 1b2 +c2 +1a2 +c2 +1a2 +b2 ) -3 ≥ 12 [(b2 +c2 )&;#183; 1b2 +c2 +(a2 +c2 )&;#183; 1a2 +c2 +(a2 +b2 )&;#183; 1a...
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