Chứng minh
a) 4cos150cos210cos240−cos120−cos180
=1+√32;
b) tan300+tan400+tan500+tan600
=8√33cos200;
c) 1sin180−1sin540=2;
d) tan90−tan270−tan630+tan810=4.
a)
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4cos150cos210cos240−cos120−cos180=2cos150(cos450+cos30)−2cos150cos30=2cos150cos450=cos600+cos300=12+√32.
b)
tan300+tan400+tan500+tan600=sin900cos300cos600+sin900cos400cos500=cos900+cos100+cos900+cos30012cos100cos300=4cos200cos100cos100cos300=8√3cos200=8√33cos200.
c)
1sin180−1sin540=sin540−sin180sin180sin540=2cos360sin180sin180sin540=2cos360sin540=2cos360cos360=2.
d)
tan90−tan270−tan630+tan810=tan90+tan810−(tan270+tan630)=(sin90cos90+sin810cos810)−(sin270cos270+sin630cos630)=1sin90cos90−1sin270cos270=2sin180−2sin540=2.2=4