Question

If A,B,C and D are the successive angles of a cyclic quadrilateral, then cosA+cosB+cosC+cosD=?

Answer 1

Cyclic quadrilateral is a quadrilateral whose all vertices lie on single circle.
In cyclic quadrilateral the sum of opposite angles is 180.

Now,
cosA + cosC = 2cos (180/2) cos(A-C/2)
cosA + cosC = 2cos (180/2) cos(A-C/2)
cosA + cosC = 0 (as cos 90 = 0)
cosB + cosD = 2cos ( B+D/2 ) cos (B-D/2)
cosB + cosD = 2cos 90 cos (B-D/2)
cosB + cosD = 0
Therefore,
cos A + cosB + cosC + cosD=0