Question

[Mai Anh]

Giải phương trình:

\(Sin^2\left(x-\dfrac{\pi}{4}\right)=Cos^2x\)

Answer 1

[Nguyễn Hoàng Minh]

\(PT\Leftrightarrow\sin^2\left(x-\dfrac{\pi}{4}\right)=\sin^2\left(\dfrac{\pi}{2}-x\right)\\ \Leftrightarrow\left|\sin^2\left(x-\dfrac{\pi}{4}\right)\right|=\left|\sin^2\left(\dfrac{\pi}{2}-x\right)\right|\Leftrightarrow\left[{}\begin{matrix}\sin^2\left(x-\dfrac{\pi}{4}\right)=\sin^2\left(\dfrac{\pi}{2}-x\right)\left(1\right)\\\sin^2\left(x-\dfrac{\pi}{4}\right)=-\sin^2\left(\dfrac{\pi}{2}-x\right)\left(2\right)\end{matrix}\right.\\ \left(1\right)\Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{4}=\dfrac{\pi}{2}-x+k2\pi\\x-\dfrac{\pi}{4}=\pi-\dfrac{\pi}{2}+x+k2\pi\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{3\pi}{8}+k\pi\\x\in\varnothing\end{matrix}\right.\\ \Leftrightarrow x=\dfrac{3\pi}{8}+k\pi\left(k\in Z\right)\)

\(\left(2\right)\Leftrightarrow\sin\left(x-\dfrac{\pi}{4}\right)=\sin\left(x-\dfrac{\pi}{2}\right)\\ \Leftrightarrow\left[{}\begin{matrix}x-\dfrac{\pi}{4}=x-\dfrac{\pi}{2}+k2\pi\\x-\dfrac{\pi}{4}=\pi-x+\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x\in\varnothing\\x=\dfrac{7\pi}{8}+k\pi\end{matrix}\right.\left(k\in Z\right)\\ \Leftrightarrow S=\left\{\dfrac{3\pi}{8}+k\pi;\dfrac{7\pi}{8}+k\pi\right\}\)

Answer 2

[Ngô Thành Chung]

\(\dfrac{1-cos\left(2x-\dfrac{\pi}{2}\right)}{2}=\dfrac{1+cos2x}{2}\)

⇔ 1 - sin2x = cos2x

⇔ sin2x + cos2x = 1

⇔ \(\sqrt{2}sin\left(2x+\dfrac{\pi}{4}\right)=1\)

⇔ \(sin\left(2x+\dfrac{\pi}{4}\right)=\dfrac{\sqrt{2}}{2}\)