Certainly! Let's solve the equation step-by-step:
Given equation:
[tex]\[
(-1)^2 = (\sec x - 1)(\sec x + 1)
\][/tex]
First, let's simplify the left-hand side of the equation:
[tex]\[
(-1)^2 = 1
\][/tex]
So the equation now is:
[tex]\[
1 = (\sec x - 1)(\sec x + 1)
\][/tex]
Next, we simplify the right-hand side of the equation using the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:
[tex]\[
(\sec x - 1)(\sec x + 1) = (\sec^2 x - 1)
\][/tex]
Thus, the equation simplifies to:
[tex]\[
1 = \sec^2 x - 1
\][/tex]
Now, let's isolate [tex]\(\sec^2 x\)[/tex]:
[tex]\[
1 + 1 = \sec^2 x
\][/tex]
[tex]\[
2 = \sec^2 x
\][/tex]
To solve for [tex]\(\sec x\)[/tex], we take the square root of both sides:
[tex]\[
\sec x = \pm \sqrt{2}
\][/tex]
This implies:
[tex]\[
\sec x = \sqrt{2} \quad \text{or} \quad \sec x = -\sqrt{2}
\][/tex]
Since [tex]\(\sec x = \frac{1}{\cos x}\)[/tex], let's convert these solutions in terms of [tex]\(\cos x\)[/tex]:
[tex]\[
\cos x = \frac{1}{\sqrt{2}} \quad \text{or} \quad \cos x = -\frac{1}{\sqrt{2}}
\][/tex]
Now, convert [tex]\(\cos x\)[/tex] solutions to [tex]\(\tan x\)[/tex]:
[tex]\[
\cos x = \frac{1}{\sqrt{2}} \implies x = \pm \frac{\pi}{4} + 2k\pi \quad \text{(for integer k)}
\][/tex]
[tex]\[
\cos x = -\frac{1}{\sqrt{2}} \implies x = \pm \frac{3\pi}{4} + 2k\pi \quad \text{(for integer k)}
\][/tex]
For these angles,
[tex]\[
\tan\left(\frac{\pi}{4}\right) = 1 \quad \text{and} \quad \tan\left(\frac{3\pi}{4}\right) = -1
\][/tex]
[tex]\[
\tan\left(-\frac{\pi}{4}\right) = -1 \quad \text{and} \quad \tan\left(-\frac{3\pi}{4}\right) = 1
\][/tex]
Then we square the [tex]\(\tan\)[/tex] value for each solution:
[tex]\[
\tan^2\left(\frac{\pi}{4}\right) = 1^2 = 1
\][/tex]
[tex]\[
\tan^2\left(\frac{3\pi}{4}\right) = (-1)^2 = 1
\][/tex]
[tex]\[
\tan^2\left(-\frac{\pi}{4}\right) = (-1)^2 = 1
\][/tex]
[tex]\[
\tan^2\left(-\frac{3\pi}{4}\right) = 1^2 = 1
\][/tex]
So, we find that:
[tex]\[
\tan(x)^2 = 1
\][/tex]
Combining everything together, we notice that there are no new integer solutions for x aside from these angles. There might be no exact values fitting due to underlying constraints.
Therefore, the final solutions in terms of the primary part of the functions and simplifications would be:
[tex]\[
\tan(x)^2 = 1
\][/tex]
Upon reconciling these steps from provided clean calculations, we infer the overall results tally unite within unwary boundaries of our functions.
