Question

Biết sin x + cos x = m. Tìm \(\left|\sin^4x-\cos^4x\right|\) . Chứng minh rằng \(\left|m\right|\le\sqrt{2}\)

Answer 1

\(A=\left|\sin^4x-\cos^4x\right|=\left|\left(\sin^2x\right)^2-\left(\cos^2x\right)^2\right|\)

\(A=\left|\left(1-\cos^2x\right)^2-\left(\cos^2x\right)^2\right|=\left|1-2\cos^2x+\cos^4x-\cos^4x\right|\)

\(=\left|1-2\cos^2x\right|=\left|\sin^2x-\cos^2x\right|=\left|\left(\sin x-\cos x\right)\left(\sin x+\cos x\right)\right|\)

\(\sin x+\cos x=m\Rightarrow\cos x=m-\sin x\Rightarrow\sin x-\cos x=\sin x-m+\sin x=2\sin x-m\)

Có \(\sin x+\cos x=m\Rightarrow\sin^2x+\cos^2x+2\sin x.\cos x=m^2\)

\(\Leftrightarrow2\sin x.\cos x=m^2-1\)

\(\left(\sin x-\cos x\right)^2=\sin^2x+\cos^2x-2\sin x.\cos x=1-2.\left(m^2-1\right)=1-2m^2+2=3-2m^2\)

\(\Rightarrow\sin x-\cos x=\sqrt{\left(\sin x-\cos x\right)^2}=\sqrt{3-2m^2}\)

\(A=\left|m\sqrt{3-2m^2}\right|=\left|m\right|.\left|\sqrt{3-2m^2}\right|\)

P/s: lm đc mỗi đến đây thui à, cái CM kia chịu nhoa :)

Answer 2

\(\left(sinx+cosx\right)^2=m^2\Rightarrow1+2sinx.cosx=m^2\)\(\Rightarrow2sinx.cosx=m^2-1\)

\(\Rightarrow\left(sinx-cosx\right)^2=\left(sinx+cosx\right)^2-4sinx.cosx=m^2-2\left(m^2-1\right)=2-m^2\)

Mà \(\left(sinx-cosx\right)^2\ge0\) \(\forall x\Rightarrow2-m^2\ge0\Rightarrow m^2\le2\Rightarrow\left|m\right|\le\sqrt{2}\)

Ta lại có \(\left(sinx-cosx\right)^2=2-m^2\Rightarrow\left|sinx-cosx\right|=\sqrt{2-m^2}\)

\(A=\left|sin^4x-cos^4x\right|=\left|\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)\right|\)

\(=\left|\left(sinx-cosx\right)\left(sinx+cosx\right)\right|\)

\(=\left|m\sqrt{2-m^2}\right|=\left|m\right|\sqrt{2-m^2}\)