Solve tan^2 theta = -tan theta for all values of x

Answer 1

tan²x = - tan x

Add tan x to both sides

tan²x + tan(x) = 0

Factor out tan(x) from both

tan x (tan x +1)=0

Set each equal to 0

tan x=0

x=0

tan x + 1=0

Subtract 1 from both sides

tan x =-1

x = 3π/4 and 7π/4

The question asks for ALL values of x

You would need to add to the 0, 3π/4 and 7π/4 a multiple of 2π