Key Concepts and Formulas

• Arithmetic Mean - Geometric Mean (AM-GM) Inequality: For non-negative real numbers $a_1, a_2, \dots, a_n$, $\frac{a_1 + a_2 + \dots + a_n}{n} \ge \sqrt[n]{a_1 a_2 \dots a_n}$. Equality holds if and only if $a_1 = a_2 = \dots = a_n$.
• Trigonometric Identities:
  • $\sin^2 x + \cos^2 x = 1$
  • $\cos(A+B) - \cos(A-B) = -2 \sin A \sin B$
• Properties of Trigonometric Functions: Understanding the signs and values of $\sin x$ and $\cos x$ in the interval $[0, \pi]$.

Step-by-Step Solution

Step 1: Analyze the Given Equation and Prepare for AM-GM The given equation is $\sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta$, with $\alpha, \beta \in [0, \pi]$. We observe that the left side is a sum of terms, and the right side involves a product. This suggests the application of the AM-GM inequality. To apply AM-GM, we need non-negative terms. $\sin^4 \alpha \ge 0$, $4 \cos^4 \beta \ge 0$, and $2 > 0$. The constant term '2' can be split into $1+1$ to facilitate the application of AM-GM with four terms. Let the four non-negative terms be $a_1 = \sin^4 \alpha$, $a_2 = 4 \cos^4 \beta$, $a_3 = 1$, and $a_4 = 1$.

Step 2: Apply the AM-GM Inequality Applying the AM-GM inequality to these four terms: $\frac{\sin^4 \alpha + 4 \cos^4 \beta + 1 + 1}{4} \ge \sqrt[4]{\sin^4 \alpha \cdot 4 \cos^4 \beta \cdot 1 \cdot 1}$ Simplifying the inequality: $\frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{4} \ge \sqrt[4]{4 \sin^4 \alpha \cos^4 \beta}$ $\frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{4} \ge \sqrt{2} \sqrt[4]{\sin^4 \alpha \cos^4 \beta}$ Since $\sqrt[4]{x^4} = |x|$, we have: $\frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{4} \ge \sqrt{2} |\sin \alpha \cos \beta|$ Multiplying both sides by 4: $\sin^4 \alpha + 4 \cos^4 \beta + 2 \ge 4\sqrt{2} |\sin \alpha \cos \beta|$

Step 3: Utilize the Equality Condition of AM-GM We are given that $\sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta$. Comparing this with the result from AM-GM, $\sin^4 \alpha + 4 \cos^4 \beta + 2 \ge 4\sqrt{2} |\sin \alpha \cos \beta|$, we see that the equality must hold in the AM-GM inequality. For equality to hold, two conditions must be met:

1. All the terms in the AM-GM inequality must be equal: $\sin^4 \alpha = 4 \cos^4 \beta = 1$.
2. The term $\sin \alpha \cos \beta$ must be non-negative, so that $|\sin \alpha \cos \beta| = \sin \alpha \cos \beta$. This is because the left side of the given equation is a sum of non-negative terms and thus non-negative, implying the right side $4\sqrt{2} \sin \alpha \cos \beta$ must also be non-negative.

Step 4: Solve for $\alpha$ and $\beta$ From the equality condition $\sin^4 \alpha = 1$: Since $\alpha \in [0, \pi]$, $\sin \alpha \ge 0$. Thus, $\sin \alpha = 1$. For $\sin \alpha = 1$ in the interval $[0, \pi]$, we get $\alpha = \frac{\pi}{2}$.

From the equality condition $4 \cos^4 \beta = 1$: $\cos^4 \beta = \frac{1}{4}$. Taking the square root of both sides, $\cos^2 \beta = \sqrt{\frac{1}{4}} = \frac{1}{2}$ (since $\cos^2 \beta \ge 0$). So, $\cos \beta = \pm \frac{1}{\sqrt{2}}$.

Now, we use the condition $\sin \alpha \cos \beta \ge 0$. Since $\sin \alpha = 1$ (which is positive), we must have $\cos \beta \ge 0$. Therefore, $\cos \beta = \frac{1}{\sqrt{2}}$. For $\cos \beta = \frac{1}{\sqrt{2}}$ in the interval $[0, \pi]$, we get $\beta = \frac{\pi}{4}$.

Step 5: Calculate the Required Expression We need to find the value of $\cos(\alpha + \beta) - \cos(\alpha - \beta)$. Using the trigonometric identity $\cos(A+B) - \cos(A-B) = -2 \sin A \sin B$: Let $A = \alpha$ and $B = \beta$. Then, $\cos(\alpha + \beta) - \cos(\alpha - \beta) = -2 \sin \alpha \sin \beta$.

Substitute the values of $\alpha$ and $\beta$ we found: $\alpha = \frac{\pi}{2} \implies \sin \alpha = \sin \frac{\pi}{2} = 1$. $\beta = \frac{\pi}{4} \implies \sin \beta = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$.

Therefore, $\cos(\alpha + \beta) - \cos(\alpha - \beta) = -2 \cdot (1) \cdot \left(\frac{1}{\sqrt{2}}\right) = -\frac{2}{\sqrt{2}} = -\sqrt{2}$.

Common Mistakes & Tips

• Incorrect AM-GM Term Selection: Splitting the constant term '2' into '1+1' is crucial for the geometric mean to simplify correctly.
• Ignoring Equality Conditions: The equality condition of AM-GM is not just about terms being equal; it also implies that the expression on the right side of the given equation must be consistent with the inequality (i.e., non-negative in this case).
• Domain Issues: Carefully use the given domain $[0, \pi]$ for $\alpha$ and $\beta$ to determine the correct signs and values of their trigonometric functions. For example, $\sin \alpha = \pm 1$ becomes $\sin \alpha = 1$ due to $\alpha \in [0, \pi]$.
• Absolute Value in Geometric Mean: Remember that $\sqrt[4]{x^4} = |x|$. The context of the problem (given equality) helps resolve the absolute value.

Summary

This problem is elegantly solved using the AM-GM inequality. By applying AM-GM to carefully chosen terms, we establish an inequality that, when compared to the given equation, forces the equality condition of AM-GM to hold. This equality condition allows us to determine the specific values of $\alpha$ and $\beta$. Finally, a standard trigonometric identity is used to compute the required expression with the found values of $\alpha$ and $\beta$.

The final answer is $\boxed{-\sqrt{2}}$.