i am having trouble remember how to solve this problem without a calculator any words of wisdom would be greatly appreciated
The answer ought to be 1, since
$\displaystyle 35^o+55^o=90^o$
Therefore, viewed from within a right-angled triangle
$\displaystyle cos35^o=sin55^o$
$\displaystyle cos55^o=sin35^o$
so
$\displaystyle cos35^osin55^o+sin35^ocos55^o=cos^235^o+sin^235^o= sin^255^o+cos^255^0=1$