Chắc đề đúng là: \(\sqrt{4^ncos^{4n}x+3}+\sqrt{4^nsin^{4n}x+3}=4\)
Ta có:
\(VT\ge\sqrt{\left(2^nsin^{2n}x+2^ncos^{2n}x\right)^2+12}\)
\(VT\ge\sqrt{4^n\left(sin^{2n}x+cos^{2n}x\right)^2+12}\)
Mặt khác, áp dụng BĐT \(a^n+b^n\ge2\left(\dfrac{a+b}{2}\right)^n\)
Ta có: \(\left(sin^2x\right)^n+\left(cos^2x\right)^n\ge2\left(\dfrac{sin^2x+cos^2x}{2}\right)^n=\dfrac{2}{2^n}\)
\(\Rightarrow\left(sin^{2n}x+cos^{2n}x\right)^2\ge\dfrac{4}{4^n}\)
\(\Rightarrow VT\ge\sqrt{4^n.\dfrac{4}{4^n}+12}=4\)
Dấu "=" xảy ra khi và chỉ khi \(sin^2x=cos^2x\Leftrightarrow cos2x=0\Leftrightarrow...\)