(1)
sinα-3cosα=0
sinα=3cosα
tanα=sinα/cosα=3
1/(1-sinα)+ 1/(1+sinα)
=(1+sinα+1-sinα)/[(1-sinα)(1+sinα)]
=2/(1-sin²α)
=2(sin²α+cos²α)/cos²α
=2(tan²α +1)
=2·(3²+1)
=20
(2)
-π/2<α<0，sinα<0，cosα>0
sinα+cosα=1/5>0，-π/4<α<0
-1<tanα<0
(sinα+cosα)²/(sin²α+cos²α)=(1/5)²/1
(sin²α+cos²α+2sinαcosα)/(sin²α+cos²α)=1/25
(tan²α+1+2tanα)/(tan²α+1)=1/25
12tan²α+25tanα+12=0
(3tanα+4)(4tanα+3)=0
tanα=-4/3(舍去)或tanα=-¾
tanα=-¾
(3)
[(1-2sinαcosα)/(cos²α-sin²α)]·[(1+2sinαcosα)/(2sin²α-1)]
=[(1-2sinαcosα)(1+2sinαcosα)]/[(cos²α-sin²α)(2sin²α-1)]
=[(sinα-cosα)²(sinα+cosα)²]/[cos2α·(-cos2α)]
=[(sinα-cosα)(sinα+cosα)]²/(-cos²2α)
=(sin²α-cos²α)²/(-cos²2α)
=(-cos2α)²/(-cos²2α)
=cos²2α/(-cos²2α)
=-1
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