Cos76°cos31°+cos14°cos59°

1) $cos( \alpha )=sin( \frac{\pi}{2}- \alpha )$
2) $sin( \alpha )=cos( \frac{\pi}{2}- \alpha )$
3) $sin( \alpha + \beta )=sin( \alpha )cos( \beta )+cos( \alpha )sin( \beta )$

$cos(76^0)=sin(90^0-76^0)=sin(14^0)$
$cos(59^0)=sin(90^0-59^0)=sin(31^0)$

$cos(76^0)cos(31^0)+cos(14^0)cos(59^0)=$
$=sin(14^0)cos(31^0)+cos(14^0)sin(31^0)=sin(14^0+31^0)=$
$=sin(45^0)= \frac{ \sqrt{2} }{2}$