Find all solutions of the equation in the interval 0 2π)

Solution

Apply the identity sec
2
x=1+tan
2
x
1+tan
2
x+6tanx+4=0
tan
2
x+6tanx+5=0
Let t=tanx.
t
2
+6t+5=0
(t+5)(t+1)=0
t=−5 and −1
tanx=−5 and tanx=−1
x=π−arctan(5),2π−arctan(5),
4
3π
​
,
4
7π
​