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PT $\Leftrightarrow (\sin x +\cos x)(\sin^2 x +\cos^2 x-\sin x \cos x)=1-2\sin^2 x$$\Leftrightarrow (\sin x +\cos x)(1-\sin x \cos x)=\cos^2 x - \sin^2 x$$\Leftrightarrow (\sin x +\cos x)(1-\sin x \cos x)=(\sin x +\cos x)(-\sin x +\cos x)$$\Leftrightarrow \left[ {\begin{matrix} \sin x +\cos x=0\\1-\sin x \cos x+\sin x - \cos x=0 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} \sin (x+ \frac{\pi}{4})=0\\(1+ \sin x)(1-\cos x)=0 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} \sin (x+ \frac{\pi}{4})=0\\\sin x=-1\\\cos x=1 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} x=- \frac{\pi}{4}+k\pi\\ x=- \frac{\pi}{2}+k2\pi\\x=k2\pi \end{matrix}} \right. (k \in \mathbb{Z}).$

PT $\Leftrightarrow (\sin x +\cos x)(\sin^2 x +\cos^2 x-\sin x \cos x)=1-2\sin^2 x$$\Leftrightarrow (\sin x +\cos x)(1-\sin x \cos x)=\cos^2 x - \sin^2 x$$\Leftrightarrow (\sin x +\cos x)(1-\sin x \cos x)=(\sin x +\cos x)(-\sin x +\cos x)$$\Leftrightarrow \left[ {\begin{matrix} \sin x +\cos x=0\\1-\sin x \cos x+\sin x - \cos x=0 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} \sin (x+ \frac{\pi}{4})=0\\(1+ \sin x)(1-\cos x)=0 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} \sin (x+ \frac{\pi}{4})=0\\\sin x=-1\\\cos x=1 \end{matrix}} \right.$$\Leftrightarrow \left[ {\begin{matrix} x=- \frac{\pi}{4}+k\pi\\ x=- \frac{\pi}{2}+k2\pi\\x=k2\pi \end{matrix}} \right. (k \in \mathbb{Z}).$