Hi!

First of all, before you start solving any equation, you have to specify the set, within which you may search for solutions (the domain). In this case, it is said to be (0, 2pi) but you've got a tangent in this equation, and it's undefined for some values from the set (pi).

So the domain is (0, 2pi) \ {pi} (excluding pi).

Everything seems to be correct until this line;

2cos^2 Θ + 2cos Θ = 0

Why don't you extract 2cosΘ from brackets and divide by 2 like this:

cosΘ(cosΘ + 1) = 0

Now at least one of the factors must equal 0, so either cosΘ=0 or cosΘ=-1. But if cosΘ=-1 then Θ=pi, but pi was excluded from the domain. So the answer is, cosΘ=0

//Edit: Looking at the post below mine, I found a mistake in my solution... Of course you must also have the secant defined, so cosΘ actually can't equal 0 because it's the denominator if you consider secΘ=1/cosΘ. Sorry for misleading you.