Question

chứng minh rằng :

sin⁴x + cos⁴x = 1/4 . cos4x + 3/4

Answer 1

\(\sin^4x+\cos^4x=\dfrac{\cos4x+3}{4}\)

\(\Leftrightarrow\left(\sin^2x+\cos^2x\right)^2-2\sin^2x.\cos^2x=\dfrac{\cos4x+3}{4}\)

\(\Leftrightarrow\dfrac{1-\cos4x}{4}=2\sin^2x.\cos^2x\)

\(\Leftrightarrow\dfrac{1-\cos4x}{2}=\left(2\sin x.\cos x\right)^2\)

\(\Leftrightarrow2\sin^22x=\sin^22x\)

\(\Leftrightarrow\left[{}\begin{matrix}\sin2x=0\\\sin2x=\dfrac{1}{2}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\kappa\pi}{2}\\x=\dfrac{\pi}{12}+\kappa\pi\left(\kappa\in Z\right)\\x=\dfrac{5\pi}{12}+\kappa\pi\end{matrix}\right.\)