tan2(x) = −1 + 2 tan(−x) Find all exact solutions on [0, 2π)

Answer 1

The key to this problem is realizing that tan(-x)= -tan(x)

tan^2x= -1+2tan(-x) now make the above substitution

tan^2x= -1-2tanx

tan^2x +2tanx +1=0

(tanx+1)(tanx+1)=0

tanx=-1 which occurs at 3pi/4 and 7pi/4