<u><em>Note: It's not clear if the expressions involving '2x' actually mean squaring the tangent/secant or doubling the angle x. I'm posting the answer assuming the latest approach.</em></u>
Answer:

Step-by-step explanation:
<u>Trigonometric Equations
</u>
It's a type of equations where the variable is the argument of some of the trigonometric functions. It's generally restricted to a given domain, so the solution must be iteratively selected within all the possible answers.
The equation to solve is

Rearranging

Factoring


We come up with two different equations:
![\displaystyle \left\{\begin{matrix}tan2x+2=0....[eq1]\\sec2x-1=0....[eq2]\end{matrix}\right.](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%5C%7B%5Cbegin%7Bmatrix%7Dtan2x%2B2%3D0....%5Beq1%5D%5C%5Csec2x-1%3D0....%5Beq2%5D%5Cend%7Bmatrix%7D%5Cright.)
Let's take eq 1:

Solving for 2x

There are two sets of possible solutions:


We get two solutions
.
The first solution is out of the range
, so it's discarded

.
.
Both solutions are feasible

.
.
Only the first solution lies in the given domain. We won't take negative values of k since it will provide negative values of x and they are not allowed in the solution
Now we solve eq 2:

This leads to the solution
Or equivalently
For k=0, x=0. This solution is valid
For k=1,
. This is also valid
For k=2,
. This solution is out of range
The whole set of solutions is
