We are tasked with solving the equation:

$\cos^2(x) - 4\cos(x) - 1 = 0$

Step 1: Let $y = \cos(x)$

This transforms the equation into a quadratic equation:

$y^2 - 4y - 1 = 0$

Step 2: Solve the quadratic equation

To solve this quadratic equation, we use the quadratic formula:

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

Here, $a = 1$, $b = -4$, and $c = -1$. Plugging these values into the quadratic formula:

$y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-1)}}{2(1)}$

$y = \frac{4 \pm \sqrt{16 + 4}}{2}$

$y = \frac{4 \pm \sqrt{20}}{2}$

$y = \frac{4 \pm 2\sqrt{5}}{2}$

$y = 2 \pm \sqrt{5}$

Thus, we have two possible values for $y$:

$y_1 = 2 + \sqrt{5}, \quad y_2 = 2 - \sqrt{5}$

Step 3: Solve for $x$

Since $y = \cos(x)$, we now solve for $x$ by taking the inverse cosine of both values. However, we must check whether these values are valid, as $\cos(x)$ is only defined in the range $[-1, 1]$.

• For $y_1 = 2 + \sqrt{5} \approx 4.236$, this value is outside the domain of the cosine function, as $\cos(x)$ must lie between $-1$ and $1$. Therefore, this solution is not valid.

• For $y_2 = 2 - \sqrt{5} \approx -0.236$, this value is within the domain of the cosine function, so we can proceed.

Step 4: Find the angles

We now solve:

$\cos(x) = 2 - \sqrt{5} \approx -0.236$

Taking the inverse cosine:

$x = \cos^{-1}(-0.236)$

Using a calculator:

$x \approx 1.803 \text{ radians}$

Since cosine is periodic with a period of $2\pi$, we have the general solution:

$x = \pm 1.803 + 2k\pi \quad \text{for} \quad k \in \mathbb{Z}$

Final Answer:

The solution to the equation $\cos^2(x) - 4\cos(x) - 1 = 0$ is:

$x \approx 1.803 + 2k\pi \quad \text{or} \quad x \approx -1.803 + 2k\pi \quad \text{for} \quad k \in \mathbb{Z}$

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Would you like a more detailed explanation of any step? Or do you have other questions on this topic?

Related Questions:

1. How do you solve a quadratic equation using factoring?
2. What is the cosine inverse function, and how does it work?
3. Can you solve equations with trigonometric identities without using a calculator?
4. How can you determine the general solution for trigonometric equations?
5. What is the period of the cosine function?

Tip: Always remember to check if the solutions to trigonometric equations fall within the valid range for the function (for cosine, between -1 and 1).