[Math] How to find the greatest value of this expression

Question

"This expression" is

$$\frac{5\sin^2\alpha+4\cos^2\alpha}{4\cos^2\beta+5\sin^2\beta}.$$
The answer is $1.25$

I used simple steps to simplify this, but couldn't find the greatest value, since it has $2$ kinds of angles. So how to find that?

Answer 1

$$\frac{5\sin^2\alpha+4\cos^2\alpha}{4\cos^2\beta+5\sin^2\beta}=\frac{4+\sin^2\alpha}{4+\sin^2\beta}\leq\frac{4+1}{4}=\frac{5}{4}.$$ The equality occurs for $\sin\alpha=1$ and $\sin\beta=0$, which says that the answer is $\frac{5}{4}.$

Done!