If ^4+4^4+2 = 4√2, ,[0,], then (+)-(-) is equal to:

Question

If $\sin^4\alpha+4\cos^4\beta+2 = 4\sqrt2\sin\alpha\cos\beta$, $\alpha,\beta\in[0,\pi]$, then $\cos(\alpha+\beta)-\cos(\alpha-\beta)$ is equal to:

Accepted Answer

By AM--GM on $\sin^4\alpha, 4\cos^4\beta, 1, 1$:

$$\sin^4\alpha+4\cos^4\beta+1+1 \ge 4\sqrt2\,|\sin\alpha||\cos\beta|$$

with equality iff $\sin^4\alpha=1$ and $4\cos^4\beta=1$, giving $\alpha=\dfrac\pi2$, $\beta=\dfrac\pi4$.

$$\cos(\alpha+\beta)-\cos(\alpha-\beta) = \cos\left(\tfrac\pi2+\beta\right)-\cos\left(\tfrac\pi2-\beta\right) = -\sin\beta-\sin\beta = -2\sin\tfrac\pi4 = -\sqrt2$$

Solutions · 1