2014/110) Find closed form of

 $\cos\,1^\circ \cos\,2^\circ + \cos\,2^\circ \cos\,3^\circ\cdots \cos\,88^\circ \cos\,89^\circ$

we have $2 \cos(x) \cos (y) = \cos (x+y) + \cos (y-x)$

so $2 (\cos\,1^\circ \cos\,2^\circ ) = \cos\,3^\circ + \cos\,1^\circ$
 $2 (\cos\,2^\circ \cos\,3^\circ ) = \cos\,5^\circ + \cos\,1^\circ$
so on till
$2 (\cos\,44^\circ \cos\,45^\circ ) = \cos\,89^\circ + \cos\,1^\circ$

 $2 (\cos\,45^\circ\cos\,46^\circ ) = \cos\,91^\circ + \cos\,1^\circ$

or  $2 (\cos\,45^\circ \cos\,46^\circ ) = - \cos\,89^\circ + \cos\,1^\circ$


so on till

$2 (\cos\,88^\circ \cos\,89^\circ ) = - \cos\,3^\circ + \cos\,1^\circ$

on adding above for each positive term in 1st half for the first term there is a -ve term for the second half and we are left wth 88 times $\cos\,1^\circ$

so $2( \cos\,1^\circ \cos\,2^\circ + (\cos\,2^\circ \cos\,3^\circ +\cdots \cos\,88^\circ \cos\,89^\circ)  = 88 \cos\,1^\circ$

or  $( \cos\,1^\circ \cos\,2^\circ + (\cos\,2^\circ \cos\,3^\circ +\cdots \cos\,88^\circ \cos\,89^\circ)  = 44 \cos\,1^\circ$