\(\Leftrightarrow\left(sin^2\dfrac{x}{2}+cos^2\dfrac{x}{2}+2sin\dfrac{x}{2}.cos\dfrac{x}{2}\right)^2=cos^2\dfrac{x}{2}-sin^2\dfrac{x}{2}\)
\(\Leftrightarrow\left(sin\dfrac{x}{2}+cos\dfrac{x}{2}\right)^4=\left(cos\dfrac{x}{2}-sin\dfrac{x}{2}\right)\left(cos\dfrac{x}{2}+sin\dfrac{x}{2}\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}cos\dfrac{x}{2}+sin\dfrac{x}{2}=0\Leftrightarrow tan\dfrac{x}{2}=-1\Leftrightarrow...\\\left(sin\dfrac{x}{2}+cos\dfrac{x}{2}\right)^3=cos\dfrac{x}{2}-sin\dfrac{x}{2}\left(1\right)\end{matrix}\right.\)
Xét (1):
Với \(cos\dfrac{x}{2}=0\) không là nghiệm
Với \(cos\dfrac{x}{2}\ne0\) chia 2 vế cho \(cos^3\dfrac{x}{2}\)
\(\Rightarrow\left(tan\dfrac{x}{2}+1\right)^3=\dfrac{1}{cos^2\dfrac{x}{2}}-tan\dfrac{x}{2}.\dfrac{1}{cos^2\dfrac{x}{2}}\)
\(\Leftrightarrow tan^3\dfrac{x}{2}+3tan^2\dfrac{x}{2}+3tan\dfrac{x}{2}+1=1+tan^2\dfrac{x}{2}-tan\dfrac{x}{2}\left(1+tan^2\dfrac{x}{2}\right)\)
\(\Leftrightarrow2tan^3\dfrac{x}{2}+2tan^2\dfrac{x}{2}+4tan\dfrac{x}{2}=0\)
\(\Leftrightarrow2tan\dfrac{x}{2}\left(tan^2\dfrac{x}{2}+tan\dfrac{x}{2}+2\right)=0\)
\(\Leftrightarrow tan\dfrac{x}{2}=0\)
\(\Leftrightarrow...\)
