1。已知sin（α+π/3）+sinα＝-4根號3/5（-π/2<α<0）,求cosα的值。2。已知sinxcosy=tanαcotγ,sinycosx=tanβcotγ,sin(x+y)sin(x-y)=cos平方γ。求證：sec2^α-sec2^β=sin2^γ 問題補充：第二題sec2^α-sec2^β=sin2^γ，2^表示平方。第一題-4根號3/5表示（-4根號3）/5

熱心網友

1。已知sin（α+π/3）+sinα＝-(4√3)/5（-π/2<α<0）,求cosα的值。sin(α+π/3)+sinα=sinαcos(π/3)+cosαsin(π/3)+sinα =(3/2)sinα+[(√3)/2]cosα所以(3/2)sinα+[(√3)/2]cosα=-(4√3)/5即3sinα+(√3)cosα=-(8√3)/5因-π/2<α<0，有sinα=- cosα所以-3cosα+(√3)cosα=-(8√3)/5cosα=4[(√3)-1]/5 2。已知sinxcosy=tanαcotγ,sinycosx=tanβcotγ,sin(x+y)sin(x-y)=(cosγ)^2。求證：(secα)^2-(secβ)^2=(sinγ)^2證明：因sinxcosy=tanαcotγ,sinycosx=tanβcotγ 兩式相加、減分別得sin(x+y)=cotγ(tanα+tanβ)sin(x-y)=cotγ(tanα-tanβ)兩式相乘，得sin(x+y)sin(x-y)=(cotγ)^2(tanα+tanβ)(tanα-tanβ)即(cotγ)^2(tanα+tanβ)(tanα-tanβ)=(cosγ)^2(tanα+tanβ)(tanα-tanβ)=(cosγ)^2/(cotγ)^2=(sinγ)^2 而sec2^α-sec2^β=1+(tanα)^2 -[1+(tanβ)^2] =(tanα)^2-(tanβ)^2 =(tanα+tanβ)(tanα-tanβ) =(sinγ)^2 證畢。