Question

Find the exact solutions to each equation for the interval 0<=x<2pi. tan^3 x + 2tan^2 + tan x=0

Answer 1

tan x (tan^2 +2tan+1)=0
or tanx=0 or (tanx +1)=0
solve it for x

Answer 2

tanx(tan^2x+2tanx+1)=0
tanx=0 tan^2x+2tanx+1=0
tanx=0 => remember tanx=sinx/cosx (so we are looking for where sinx is zero)
so x=0, pi ,and 2pi
now we also have tan^2x+2tanx+1=0
let u=tanx
u^2+2u+1=0
(u+1)(u+1)=0
u+1=0
u=-1
but remember tanx=u
tanx=-1
sinx/cosx=-1
we are looking for where sinx and cosx are the opposite
so x=3pi/4,7pi/4