Question

if A+B+C+D=360,sothat cos1/2(A+C)+cos1/2(B+D)=?

Answer 1

if A+B+C+D=360 then
A+C=360-(B+D). therefore :
cos1/2(A+C) = cos1/2(360-(B+D))
= cos(180 - 1/2(B+D))
use the identity :
cos(180 - x) = -cos(x)
so,
cos1/2(A+C) = cos(180 - 1/2(B+D)) = -cos1/2(B+D)

now, the original question asking us cos1/2(A+C)+cos1/2(B+D).
because the value of cos1/2(A+C) = -cos1/2(B+D), then
cos1/2(A+C) + cos1/2(B+D) = -cos1/2(B+D) + cos1/2(B+D) = 0