Xét \(\dfrac{1}{\sqrt{2}}.\dfrac{1}{\sqrt{2+\sqrt{3}}}=\dfrac{1}{\sqrt{4+2\sqrt{3}}}\)
= \(\dfrac{1}{\sqrt{\left(1+\sqrt{3}\right)^2}}=\dfrac{1}{1+\sqrt{3}}\)
<=> \(\dfrac{1}{\sqrt{2+\sqrt{3}}}=\dfrac{\sqrt{2}}{1+\sqrt{3}}\)
Xét \(\sqrt{2}\sqrt{2+\sqrt{3}}=\sqrt{4+2\sqrt{3}}=\sqrt{\left(1+\sqrt{3}\right)^2}=1+\sqrt{3}\)
<=> \(\sqrt{2+\sqrt{3}}=\dfrac{1+\sqrt{3}}{\sqrt{2}}\)
<=> P = \(\dfrac{\sqrt{2}}{1+\sqrt{3}}-\dfrac{1+\sqrt{3}}{\sqrt{2}}\)
= \(\dfrac{2-\left(1+\sqrt{3}\right)^2}{\sqrt{2}\left(1+\sqrt{3}\right)}=\dfrac{2-1-3-2\sqrt{3}}{\sqrt{2}\left(1+\sqrt{3}\right)}=\dfrac{-2-2\sqrt{3}}{\sqrt{2}\left(1+\sqrt{3}\right)}=\dfrac{-2}{\sqrt{2}}=-\sqrt{2}\)
