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证1/(cos0cos1)+1/(cos1cos2)+1/(cos2cos3)+.+1/(cos88cos89)=cos1/(sin1)^2
题目详情
证1/(cos0cos1)+1/(cos1cos2)+1/(cos2cos3)+.+1/(cos88cos89)=cos1/(sin1)^2
▼优质解答
答案和解析
cos(a-b)/(cosacosb)=(cosacosb+sinasinb)/(cosacosb)=1+tanatanb.
tan(a-b)=(tana-tanb)/(1+tanatanb)
--->1+tanatanb=(tana-tanb)/tan(a-b)
1/(cos0cos1)+1/(cos1cos2)+.+1/(cos88cos89)
=1/cos1*[cos1/(cos0cos1)+.]
=1/(cos1)*[cos(1-0)/(cos0cos1)+.]
=1/cos1)*[(1+tan0tan1)+(1+tan1tan2)+.]
=1/cos1*[(tan1-tan0)/tan1+(tan2-tan1)/tan1+.+(tan89-tan88)/tan1]
=1/cos1*tan89/tan1
=1/cos1*cot1/tan1
=1/cos1*(cos1/sin1)^2
=cos1/(sin1)^2
tan(a-b)=(tana-tanb)/(1+tanatanb)
--->1+tanatanb=(tana-tanb)/tan(a-b)
1/(cos0cos1)+1/(cos1cos2)+.+1/(cos88cos89)
=1/cos1*[cos1/(cos0cos1)+.]
=1/(cos1)*[cos(1-0)/(cos0cos1)+.]
=1/cos1)*[(1+tan0tan1)+(1+tan1tan2)+.]
=1/cos1*[(tan1-tan0)/tan1+(tan2-tan1)/tan1+.+(tan89-tan88)/tan1]
=1/cos1*tan89/tan1
=1/cos1*cot1/tan1
=1/cos1*(cos1/sin1)^2
=cos1/(sin1)^2
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