Let $x$ and $y$ be two real numbers such that $2\sin x\sin y + 3\cos y + 6\cos x\sin y = 7$. Then $\tan^2 x + \tan^2 y$ is equal to:
Step-by-Step Solution
Key Concept: Let $\vec{a}=(2,3,6)$ and $\vec{b}=(\sin x\sin y, \cos y, \cos x\sin y)$. Then $\vec{a}\cdot\vec{b}=2\sin x\sin y+3\cos y+6\cos x\sin y=7$. Also $|\vec{a}|=\sqrt{4+9+36}=7$ and $|\vec{b}|=\sqrt{\sin^2 y(\sin^2 x+\cos^2 x)+\cos^2 y}=1$. So $\vec{a}\cdot\vec{b}=7=|\vec{a}||\vec{b}|$, meaning $\vec{a}\parallel\vec{b}$.
$\tan^2 x+\tan^2 y=\frac{1}{9}+\frac{40}{9}=\frac{41}{9}\approx\mathbf{4.56}$.
Correct Answer: 4.56