To solve the equation
[tex]\[
\cos \left(x^2 + y^2 + z^2 - xy - yz - zx\right)^0 + \sin \left(x^2 + y^2 + z^2 + 2xy^2 + 2yz + 2xz + 90\right)^2 = 2,
\][/tex]
we will walk through a series of logical steps.
1. Simplify the terms with exponents:
- [tex]\(\cos\left(\text{expression}\right)^0 = \cos\left(\text{anything}\right)^0\)[/tex]. Any number raised to the power of 0 equals 1.
- So, [tex]\(\cos \left(x^2 + y^2 + z^2 - xy - yz - zx\right)^0\)[/tex] simplifies to [tex]\(1\)[/tex].
2. Rewrite the equation:
- Now our equation simplifies to:
[tex]\[
1 + \sin\left(x^2 + y^2 + z^2 + 2xy^2 + 2yz + 2xz + 90\right)^2 = 2
\][/tex]
3. Isolate the sine term:
- Subtract 1 from both sides:
[tex]\[
\sin\left(x^2 + y^2 + z^2 + 2xy^2 + 2yz + 2xz + 90\right)^2 = 1
\][/tex]
4. Analyze the sine term:
- For [tex]\(\sin(\text{something})^2 = 1\)[/tex], the sine function itself must be equal to [tex]\(\pm1\)[/tex].
- Recall that [tex]\(\sin(\theta) = 1\)[/tex] or [tex]\(\sin(\theta) = -1\)[/tex] at specific angles:
- [tex]\(\theta = \frac{\pi}{2} + k\pi\)[/tex], where [tex]\(k\)[/tex] is an integer.
5. Set up the equation for the angle:
- We need the entire argument of the sine function to equal an angle where the sine is equal to [tex]\(\pm1\)[/tex]. Thus:
[tex]\[
x^2 + y^2 + z^2 + 2xy^2 + 2yz + 2xz + 90 = \frac{\pi}{2} + k\pi
\][/tex]
6. Recognize that this represents an equation in three variables (x, y, z):
- Solving for exact values of [tex]\(x, y,\)[/tex] and [tex]\(z\)[/tex] can be complex and typically would require either numerical solutions or specific values that satisfy the equation.
For a simplified example, if [tex]\(x, y, z\)[/tex] are all set to 1, we observe if they satisfy:
- Substitute [tex]\(x = 1, y = 1, z = 1\)[/tex]:
- The term within the sine function becomes:
[tex]\[
1^2 + 1^2 + 1^2 + 2 \cdot 1 \cdot 1^2 + 2 \cdot 1 \cdot 1 + 2 \cdot 1 \cdot 1 + 90 = 1 + 1 + 1 + 2 + 2 + 2 + 90 = 99.
\][/tex]
7. Determine if the angle condition fits:
- We need [tex]\(99\)[/tex] such that [tex]\(\sin(99)\)[/tex] corresponds to [tex]\(\pm1\)[/tex].
- Convert the number inside the sine function to degrees to verify:
- [tex]\( 99 \equiv 90° \mod 180°\)[/tex], which means [tex]\(99\)[/tex] is an angle where [tex]\(\sin(90°) = \pm1\)[/tex].
Therefore, by plugging in [tex]\(x = 1, y = 1, z = 1\)[/tex], we see that:
[tex]\[
\sin(99)^2 = 1
\][/tex]
matches our condition. Hence, one set of solutions for [tex]\(x, y, z\)[/tex] where the equation holds true is [tex]\(x = 1, y = 1, z = 1\)[/tex].
