stropa0u

2022-01-22

Solving $\mathrm{cos}\frac{165}{2}$

terorimaox

Expert

Note that $\sqrt{2\sqrt{2}}=\sqrt{\frac{4}{\sqrt{2}}}$ so you have
$f=\frac{\sqrt{2\sqrt{2}\left(2\sqrt{2}-1-\sqrt{3}\right)}}{4}$
$=\frac{12}{\sqrt{\frac{2\sqrt{2}-1-\sqrt{3}}{\sqrt{2}}}}$
$=\frac{12}{\sqrt{2-\frac{1+\sqrt{3}}{\sqrt{2}}}}$
so remains to prove that
$\frac{1+\sqrt{3}}{\sqrt{2}}=\sqrt{2+\sqrt{3}}$
To do that,multiply by $\sqrt{2}$ and square both sides to get
${1}^{2}+3+2\sqrt{3}=2\left(2+\sqrt{3}\right)$
which are obviously equal to $4+2\sqrt{3}$

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