2 cot x + sec² x = 0

Steps to Solve

1. Rearranging the equation

We start with the equation:

$$ 2 \cot x + \sec^2 x = 0 $$

Rearranging gives us:

$$ \sec^2 x = -2 \cot x $$

2. Using trigonometric identities

Recall that $\sec^2 x = 1 + \tan^2 x$ and $\cot x = \frac{1}{\tan x}$. We can substitute $\sec^2 x$:

$$ 1 + \tan^2 x = -2 \cot x $$

This can also be expressed as:

$$ 1 + \tan^2 x = -2 \frac{1}{\tan x} $$

3. Multiplying through by $\tan x$

To eliminate the fraction, multiply both sides by $\tan x$ (assuming $\tan x \neq 0$):

$$ \tan x + \tan^3 x = -2 $$

4. Rearranging to form a polynomial

This gives us the polynomial equation:

$$ \tan^3 x + \tan x + 2 = 0 $$

5. Substituting $y = \tan x$

Let $y = \tan x$:

$$ y^3 + y + 2 = 0 $$

6. Using the Rational Root Theorem

To find rational roots, we can test possible values. Testing $y = -1$:

$$ (-1)^3 + (-1) + 2 = -1 - 1 + 2 = 0 $$

Thus, $y = -1$ is a root.

7. Factoring the polynomial

Now we can factor the polynomial using synthetic division or polynomial division. Dividing:

$$ y^3 + y + 2 = (y + 1)(y^2 - y + 2) $$

8. Solving for the remaining roots

Next, we solve $y^2 - y + 2 = 0$ using the quadratic formula:

$$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 - 8}}{2} = \frac{1 \pm \sqrt{-7}}{2} $$

This yields complex roots. We focus on:

$$ y = \tan x = -1 $$

9. Finding the angles for $\tan x = -1$

The angles where $\tan x = -1$ are $x = \frac{3\pi}{4} + n\pi$, where $n$ is any integer.