User:Satori/proof

${\frac {{\sqrt {2}}-{\sqrt {6}}}{4}}={\frac {-{\sqrt {2-{\sqrt {3}}}}}{2}}$

Multiplying by two:

${\frac {{\sqrt {2}}-{\sqrt {6}}}{2}}=-{\sqrt {2-{\sqrt {3}}}}$

Squaring both sides:

${\frac {2-2({\sqrt {2}}{\sqrt {6}})+6}{2^{2}}}=2-{\sqrt {3}}$

Simplified:

${\frac {8-2{\sqrt {12}}}{4}}=2-{\sqrt {3}}$

Multiplying by four:

$8-2{\sqrt {12}}=8-4{\sqrt {3}}$

Subtracting eight and multiplying by negative one:

$2{\sqrt {12}}=4{\sqrt {3}}$

Factoring a two from the right side and representing as the square root of four:

$2{\sqrt {12}}=2{\sqrt {4}}{\sqrt {3}}$

$2{\sqrt {12}}=2{\sqrt {12}}$