证明恒等式：tanαtan2αtan2α−tanα+3（sin2α-cos2α）=2sin（2α-π3）．

Question

证明恒等式：(tanαtan2α)/(tan2α−tanα)+√(3)（sin²α-cos²α）=2sin（2α-(π)/(3)）．

Answer 1

证明：∵左边＝((sinα)/(cosα)•(sin2α)/(cos2α))/((sin2α)/(cos2α)−(sinα)/(cosα))−√(3)cos2α=(sinαsin2α)/(sin2αcosα−cos2αsinα)−√(3)cos2α
＝(sinαsin2α)/(sin(2α−α))−√(3)cos2α＝sin2α−√(3)cos2α=2sin（2α-(π)/(3)）=右边，
所以等式成立．