Find all theta in the interval [0, 2pi) that satisfy the equation. 2 cos theta tan theta + tan theta = 0

(I'm going to replace theta with "x" for efficiency purposes)

You can factor out a tanx to get:

Tanx(2cosx+1) = 0

Now set each part equal to 0 to find answers:

Tanx=0
X= pi and 2pi

Now second part:

2cosx+1=0

2cosx = -1

Cosx= -1/2

X= 2pi/3 and 4pi/3

So your answer is:
X= pi, 2pi, 2pi/3, and 4pi/3

I hope this Helps!