Question

letA B C Dare the angles of a cyclic quadrilateral then cosA+ cosB + cosC + cosD equals

Answer 1

Cyclic quadrilateral is a quadrilateral whose all vertices lie on a single circle.

In cyclic quadrilateral, the sum of opposite angles is 180°.

i•e, A+C= 180° and B+D= 180°

cosA+cosC = 2cos(A+C/2)cos(A-C/2)

cosA+cosC = 2cos(180°/2)cos(A-C/2)

cosA+cosC= 0 {as cos90°=0}

cosB+cosD = 2cos(B+D/2)cos(B-D/2)

cosB+cosD = 2cos90°cos(B-D/2)

cosB+cosD = 0

Therefore,

cosA+cosB+cosC+cosD=0...