Question

证明tan（阿尔法+四分之派）+tan（阿尔法+四分之三派）=2tan2阿尔法

Answer 1

tan（a+π/4）+tan（a+3π/4）
=tan（a+π/4）+tan（π/2+a+π/4）
=tan（a+π/4）-cot（a+π/4）
=sin(a+π/4)/cos(a+π/4)-cos(a+π/4)/sin(a+π/4)
=(sin²(a+π/4)-cos²(a+π/4))/(sin(a+π/4)cos(a+π/4))
= -(cos(2a+π/2))/((sin(2a+π/2))/2)
=sin2a/((cos2a)/2)
=2tan²2a

追问

可以换种做法吗？ cot老师没讲过...

追答

tan（a+π/4）+tan（a+3π/4）
=(tana+tanπ/4)/(1-tana*tanπ/4)+(tana+tan3π/4)/(1-tana*tan3π/4)
=(tana+1)/(1-tana)+(tana-1)/(1+tana)
=((tana+1)²-(tana-1)²)/(1-tan²a)
=(4tana)/((cos²a-sin²a)/cos²a)
=4(sina/cosa)cos²a/cos2a
=4(sinacosa)/cos2a
=2sin2a/cos2a
=2tan2a

追问

问下哦 =(tana+1)/(1-tana)+(tana-1)/(1+tana)
怎么算的呢?