Answer
$-\dfrac {\sqrt {2}}{2}$
Work Step by Step
$2\cos ^{2}\alpha -1=\cos _{2}\alpha \Rightarrow 2\cos ^{2}67\dfrac {1}{2}-1=\cos \left( 2\times \left( 67\dfrac {1}{2}\right) \right) =\cos 135=-\dfrac {\sqrt {2}}{2}$
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