In a cyclic quadrilateral ABCD, the value of `2+sum cos A cos B,` is

Options

A. `sin^(2)A+sin^(2)B,` is
B. `sin^(2)B+sin^(2),` D
C. `sin^(2)A+sin^(2)C`
D. `sin^(2)B+sin^(2)C`

Answer

A

Explanation

In the cyclin quadrilateral ABCD, we have `A+C=pi=B+D`
`thereforecos A+cosB+cos C+cosD)^(2)=0`
`implies(cosA+cosB+cosC+cosD)^(2)=0`
`impliescos^(2)A+cosB+cosC+cos^(2)D+2sumcosAcosB=0`
`implies2 cos^(2)A+2cos^(2)B+2sumcosAcosB=0`
`" "[becausecosC=-cosA and cosD=-cosB]`
`impliessum cos A cos B=-[cos^(2)A+cos^(2)B]`
`sumcosAcosB=-2+sin^(2)A+sin^(2)B`
`implies2+sumcos A cos B=sin^(2)A+sin^(2)B=sin^(2)C+sin^(2)D.`