#### Question

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 ?

Question

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

\therefore 2\sin x\cos x\leq 1

\therefore Q\leq 2

Therefore, Q