<s>[INST] Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $ Prove that for $a,b\in [-1,1]$:

$$a\sqrt{1-b^2}+b\sqrt{1-a^2}\leq 1$$

 [/INST] By Cauchy-Schwarz
$$(a\sqrt{1-b^2}+b\sqrt{1-a^2})^2 \leq (a^2+b^2)(1-b^2+1-a^2)=2(a^2+b^2)-(a^2+b^2)^2$$
$$=1-(1-a^2-b^2)^2\leq 1$$
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