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Value of a numeric expression

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In a \mathrm{AB} \Gamma triangle it holds that

(1)   \begin{equation*} \cos^2 2\mathrm{A} + \cos^2 2 \mathrm{B} + \cos^2 2 \Gamma =1 \end{equation*}

Prove that

    \[\left ( \mathrm{A} - \frac{\pi}{4} \right ) \left ( \mathrm{A} - \frac{3 \pi}{4} \right ) \left ( \mathrm{B} - \frac{\pi}{4} \right ) \left ( \mathrm{B} - \frac{3 \pi}{4} \right ) \left ( \Gamma - \frac{\pi}{4} \right ) \left( \Gamma - \frac{3 \pi}{4} \right ) = 0\]

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