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

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