Question : Let $A, B, C,$ and $D$ be the angles of a quadrilateral. If they are concyclic, then the value of $\cos\ A+\cos\ B+\cos\ C+\cos\ D$ is:
Option 1: 0
Option 2: 1
Option 3: –1
Option 4: 2
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Correct Answer: 0
Solution :
$A, B, C$, and $D$ are the angles of a quadrilateral and they are concyclic.
So, $\angle A+\angle C=\angle B+\angle D=180°$
$\angle A=180°–\angle C$
$\therefore\cos\ A=\cos(180°-C)=–\cos\ C$
Similarly, $\cos\ D=–\cos\ B$
$\cos\ A+\cos\ B+\cos\ C+\cos\ D$
$=\cos\ A+\cos\ B-\cos\ A-\cos\ B$
$=0$
Hence, the correct answer is $0$.
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