If $\cos^4 \alpha+4\sin^4 \beta-4\sqrt{2}\cos \alpha \sin \beta +2=0$,
where $\displaystyle \alpha, \beta \in \bigg(0,\frac{\pi}{2}\bigg)$. Then value of $\alpha,\beta$ are
Try: I am trying to convert it into sum of square of quantity
like $$(\cos^2 \alpha)^2+(2\sin^2 \beta)^2-2\cdot \cos^2 \alpha \cdot 2\sin^2 \beta-4\sqrt{2}\cos \alpha \sin \beta +2+4\cos^2 \alpha \cdot \sin^2 \beta$$
$$(\cos^2 \alpha–2\sin^2 \beta)^2-4\sqrt{2}\cos \alpha \sin \beta+4\cos^2 \alpha \cdot \sin^2 \beta$$
Now i did not know how to solve it, could some help me
Best Answer
Hint:
For real $\cos^2\alpha,\sin^2\beta$
$$\dfrac{\cos^4\alpha+4\sin^4\beta+1+1}4\ge\sqrt[4]{\cos^4\alpha\cdot4\sin^4\beta}$$
The equality will occur if $\cos^4\alpha=4\sin^4\beta=1$