证明下列恒等式(1)tan(x/2+π/4)+tan(x/2-π/4)=2tanx
1个回答

1.tan(x/2+π/4)+tan(x/2-π/4)

=[tan(x/2)+tan(π/4)]/[1-tan(x/2)tan(π/4)]+[tan(x/2)-tan(π/4)]/[1+tan(x/2)tan(π/4)]

=[tan(x/2)+1]/[1-tan(x/2)]+[tan(x/2)-1]/[1+tan(x/2)]

=[(tan(x/2)+1)^2-(tan(x/2)-1)^2]/[1-(tan(x/2))^2]

=4tan(x/2)/[1-(tan(x/2))^2]

=2tanx

2.(1-2sinαcosα)/(cos²α-sin²α)

(1-tanα)/(1+tanα)

=[(cosa-sina)/cosa]/[(cosa+sina)/cosa]

=(cosa-sina)/(cosa+sina)

=(cosa-sina)²/(cos²a-sin²a)

=(1-2sinαcosα)/(cos²α-sin²α)