5.
ĐK: \(x\ne\pi+k2\pi\)
Đặt \(t=tan\dfrac{x}{2}\Rightarrow sinx=\dfrac{2t}{1+t^2}\)
Phương trình trở thành:
\(\dfrac{2t}{1+t^2}+t=2\)
\(\Leftrightarrow2t+t+t^3=2+2t^2\)
\(\Leftrightarrow t^3-2t^2+3t-2=0\)
\(\Leftrightarrow t=1\)
\(\Leftrightarrow tan\dfrac{x}{2}=1\)
\(\Leftrightarrow sin\dfrac{x}{2}=cos\dfrac{x}{2}\)
\(\Leftrightarrow\dfrac{x}{2}=\dfrac{\pi}{4}+k\pi\)
\(\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)
7.
\(\Leftrightarrow8\left(\dfrac{1+cos2x}{2}\right)^2=3+5\left(2cos^22x-1\right)\)
Đặt \(cos2x=t\) ta được:
\(8\left(\dfrac{t+1}{2}\right)^2=3+5\left(2t^2-1\right)\)
\(\Leftrightarrow2t^2-t-1=0\)
\(\Leftrightarrow\left[{}\begin{matrix}t=1\\t=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}cos2x=1\\cos2x=-\dfrac{1}{2}\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=k\pi\\x=\dfrac{\pi}{3}+k\pi\\x=-\dfrac{\pi}{3}+k\pi\end{matrix}\right.\)
