Given $$ a＞0 $$ and $$ a {=}\mathllap{/\,} 1 $$ , $$ P＝a^2+a^{-2} $$ , $$ Q＝(\sin x+\cos x)^2 $$ , which one is larger, $$ P $$ or $$ Q $$ ?

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Uniteasy Admin · Accepted Answer

According to Inequality of arithmetic and geometric means AM–GM inequality,
$$ a^2+a^{-2}>2\sqrt{a\cdotp a^{-1} }=2 $$
$$ (\sin x+\cos x)^2=1+2\sin x\cos x $$
$$ \because (\sin x-\cos x)^2 \geq0 $$
$$ 	herefore 2\sin x\cos x\leq 1 $$
$$ 	herefore Q\leq 2 $$
Therefore, Q<P

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