If sin^4 alpha + 4 cos^4 beta + 2 = 4sqrt{2} | vedclass

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(D) Using the $AM \geq GM$ inequality for the four terms $\sin^4 \alpha, 4 \cos^4 \beta, 1, 1$:$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 1 + 1}{4} \geq

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$-\sqrt{2}$

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(D) Using the $AM \geq GM$ inequality for the four terms $\sin^4 \alpha, 4 \cos^4 \beta, 1, 1$:$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 1 + 1}{4} \geq (\sin^4 \alpha \cdot 4 \cos^4 \beta \cdot 1 \cdot 1)^{1/4}$$\frac{\sin^4 \alpha + 4 \cos^4 \beta + 2}{4} \geq (4 \sin^4 \alpha \cos^4 \beta)^{1/4} = \sqrt{2} \sin \alpha \cos \beta$$\sin^4 \alpha + 4 \cos^4 \beta + 2 \geq 4\sqrt{2} \sin \alpha \cos \beta$Given the equality holds,$AM = GM$,so $\sin^4 \alpha = 4 \cos^4 \beta = 1$.Thus,$\sin^4 \alpha = 1 \Rightarrow \sin \alpha = 1$ (since $\alpha \in [0, \pi]$).And $4 \cos^4 \beta = 1$ $\Rightarrow \cos^2 \beta = 1/2$ $\Rightarrow \cos \beta = \pm 1/\sqrt{2}$.Since $\beta \in [0, \pi]$,$\sin \beta = 1/\sqrt{2}$.If $\cos \beta = 1/\sqrt{2}$,then $\alpha = \pi/2, \beta = \pi/4 \Rightarrow \cos(\alpha + \beta) = \cos(3\pi/4) = -1/\sqrt{2}$.If $\cos \beta = -1/\sqrt{2}$,then $\alpha = \pi/2, \beta = 3\pi/4 \Rightarrow \cos(\alpha + \beta) = \cos(5\pi/4) = -1/\sqrt{2}$.

If $\sin^4 \alpha + 4 \cos^4 \beta + 2 = 4\sqrt{2} \sin \alpha \cos \beta$ and $\alpha, \beta \in [0, \pi],$ then $\cos(\alpha + \beta)$ is equal to

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