<s>[INST] When is $a \space \sin^2(x) + b \space \cos^2(x) \le 1$? When is the above expression less than or equal to $1$, meaning for what values of $a$ and $b$ will the above expression be less than or equal to $1$?
 [/INST] Well, clearly it equals 1 when $a=b=1$. Rewrite $b=a+c$. Then you are asking when
$$a\sin^2 x+b\cos^2x=a\sin^2x+(a+c)\cos^2x=a(\sin^2x+\cos^2x)+c\cos^2x=a+c\cos^2x>1$$
or when
$$\cos^2x>\frac{1-a}{c}=\frac{1-a}{b-a}$$
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