Question

If A, B, C and D are the angles of a cyclic quadrilateral, prove that cosA + cosB + cosC + cosD = 0.

Answer 1

In a cyclic quad we know
A+C = 180
A= C-180

Cos A= Cos (C-180)
Cos A= -Cos C

Now,
Cos A+ Cos C
-Cos C + Cos C =0
[cosA = -cos C]

Similary with

Cos B + Cos D