Question 1036513
We'll use the trig identity 
{{{cos(X) cos(Y) + sin(X) sin (Y) = cos(X-Y)}}}



{{{cos(pi) cos(3pi/4) + sin(pi) sin (3pi/4) = cos(pi-3pi/4)}}}


{{{cos(pi) cos(3pi/4) + sin(pi) sin (3pi/4) = cos(4pi/4-3pi/4)}}} use the trig identity (see above)


{{{cos(pi) cos(3pi/4) + sin(pi) sin (3pi/4) = cos(pi/4)}}}


{{{cos(pi) cos(3pi/4) + sin(pi) sin (3pi/4) = (sqrt(2))/2}}} use the <a href="https://upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Unit_circle_angles_color.svg/2000px-Unit_circle_angles_color.svg.png">Unit Circle</a>



The final answer is {{{(sqrt(2))/2}}}