Question

If A, B, C, D are angles of a cyclic quadrilateral, prove that: cos A + cos B + cos C + cos D = 0.

Answer 1

Note: If the vertices of a quadrilateral lie on the circle then the quadrilateral is called a cyclic quadrilateral. 

In a cyclic quadrilateral sum of opposite angles are 180°.

Since A, B, C, D are angles of cyclic quadrilateral 

A + C = 180° and B + D = 180° 

LHS = cos A + cos B + cos C + cos D 

= cos A + cos B + cos(180° – A) + cos(180° – B) 

= cos A + cos B – cos A – cos B 

= 0 

= RHS