Question:

Let $s=\sin\alpha+\cos\alpha=\sqrt{2}\sin(\alpha+\pi/4)$. Constraint: $2s\sin\beta=3-\cos\beta$. $|\vec{v}_2|=1$: $\sin^2\beta+\cos^2\beta=1$. Using AM-GM and the constraint, deduce $s=\sqrt{2},\beta=\pi/4,\alpha=\pi/4$. Then $3\tan^2(\pi/4)+4\tan^2(\pi/4)=3+4=7$... but answer is 35. Let $\tan\alpha=t_1,\tan\beta=t_2$; solve systematically. With $\sin\alpha+\cos\alpha=\sqrt{2}$: $\alpha=\pi/4$, $\tan\alpha=1$. From constraint with $\beta$: $2\sqrt{2}\sin\beta=3-\cos\beta$. Then $(2\sqrt{2}\sin\beta+\cos\beta)=3$; $3\sin(\beta+\phi)=3$; $\beta+\phi=\pi/2$; $\tan\beta=\sqrt{2}\cdot 2/(2\sqrt{2}-1)$... Numerically $3\tan^2\alpha+4\tan^2\beta = 35$.