证明:tanα+2tan2α+2^2tan(2^2α)+……+2^ntan(2^nα)=cotα-2^(n+1)cot2^(n+1)α

问题描述:

证明:tanα+2tan2α+2^2tan(2^2α)+……+2^ntan(2^nα)=cotα-2^(n+1)cot2^(n+1)α

cot2^nα-tan2^nα=cos2^nα/sin2^nα-sin2^nα/cos2^nα= [(cos2^nα)²-( sin2^nα) ²]/[sin2^nαcos2^nα]= cos2^(n+1)α/[1/2 sin2^(n+1)α]=2 cos2^(n+1)α/sin2^(n+1)α=2 cot2^(n+1)α.即cot2^nα-t...