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Both answers above are incomplete.
`tanx=0` implies that `x=0+kpi` where k is an integer.(Or `k*180^@` if you are working in degrees)
`tanx=1` implies that `x=kpi + pi/4` where k is an integer. (Or `k*180^@+45^@`)
Thus the full answer is `x=kpi",or,"x=kpi+pi/4` for `k in ZZ` .
given tan^2x = tanx
thus tan^2x - tanx = 0 , (take tanx common)
tanx(tanx-1) = 0 , this shows that
either tanx =0 , or tanx-1 = 0
either x = tan^-1 (0) = pi/2, or tanx = 1
either x = pi/2 or x = tan^-1(1) = 45',
thus x= pi/2 or pi/4.
Q: find x , tan^2 x = tan x
A: tan^2 x = tan x
tan^2 x - tan x = 0
tan x( tan x - 1) = 0
So, either tan x = 0 or tan x -1 =0
either tan x = 0 or tan x = 1
but tan 0 = 0, tan 45 = 1
so x= 0 or x= 45 (pi/4)
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