Question

If $ABCD$ is a cyclic quadrilateral, then the value of $\cos A+\cos B+\cos C+\cos D$ is

• A. $1$
• B. $0$
• C. $2$
• D. None of these

Answer 1

The correct option is B $0$

$ABCD$ is a cyclic quadrilateral.

We know that opposite angles of cyclic quadrilateral are supplementary.

$\therefore$ $A+C=\pi ={180}^o$

$\Rightarrow$ $A=\pi -C$

$\Rightarrow$ $\cos A=\cos (\pi -C)$

$\Rightarrow$ $\cos A=\cos \pi .\cos C+\sin \pi .\sin C$

$\Rightarrow$ $\cos A=(-1).\cos C+0.\sin C$

$\Rightarrow$ $\cos A=-\cos C$

$\therefore$ $\cos A+\cos C=0$ ----- ( 1 )

$\therefore$ $B+D=\pi ={180}^o$

$\Rightarrow$ $B=\pi -D$

$\Rightarrow$ $\cos B=\cos (\pi -D)$

$\Rightarrow$ $\cos B=\cos \pi .\cos D+\sin \pi .\sin D$

$\Rightarrow$ $\cos B=(-1).\cos D+0.\sin D$

$\Rightarrow$ $\cos B=-\cos D$

$\therefore$ $\cos B+\cos D=0$ ---- ( 2 )

Adding ( 1 ) and ( 2 ) we get,

$\Rightarrow$ $\cos A+\cos B+\cos C+\cos D=0$