Question

Assertion :If $A,B,C,D$ are angles of a cyclic quadrilateral then $\sum \sin A=0$ Reason: If $A,B,C,D$ are angles of cyclic quadrilateral then, $\sum \cos A=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

In a cyclic quadrilateral $ABCD$, $\angle A+\angle C=\pi ;\angle B+\angle D=\pi$

$\Rightarrow \cos A=\cos (\pi -C);\cos B=\cos (\pi -D)\Rightarrow \cos A+\cos C=\cos B+\cos D=0\Rightarrow \sum \cos A=0$

$\therefore$ Reason is correct, but assertion is incorrect as each of

$A,B,C,D$ lies in the interval $(0,\pi )$ in which $\sin A,\sin B,\sin C,\sin D$ are positive.

$\sin (\pi -A)=\sin A$

$\Rightarrow \sum \sin A\ne 0$

Also we have a counter example when $ABCD$ is a rectangle, where

and $\ne 0$