化简：(1/cos0cos1+1/cos1cos2+.1/cos2010cos2011)sin1

Question

化简：(1/cos0cos1+1/cos1cos2+.1/cos2010cos2011)sin1

Answer 1

cos(a-b)/(cosacosb)=(cosacosb+sinasinb)/(cosacosb)=1+tanatanb.
tan(a-b)=(tana-tanb)/(1+tanatanb)
--->1+tanatanb=(tana-tanb)/tan(a-b)
--->cos(a-b)/(cosacosb)=(tana-tanb)/(1+tanatanb)
1/(cos0cos1)+1/(cos1cos2)+.+1/(cos2010cos2011)
=(1/cos1)·[cos1/(cos0cos1)+cos1/(cos1cos2)+.+cos1/(cos2010cos2011)]·sin1 (提取cos1)
=(1/cos1)·[cos(1-0)/(cos0cos1)+cos(2-1)/(cos1cos2)+.+cos(2011-2010)/(cos2010cos2011)]·sin1
(这就和上面推导出的式子一样了)
=(1/cos1)·[(tan1-tan0)/tan1+(tan2-tan1)/tan1+.+(tan2011-tan2010)/tan1]·sin1
=(1/cos1tan1)·[(tan1-tan0)+(tan2-tan1)+.+(tan2011-tan2010)]·sin1
=(1/sin1)·tan2011·sin1
=tan2011