Question

If $A,B,C\&D$ are the angles of a cyclic quadrilateral $ABCD,$ then value of
$\cos A+\cos B+\cos C+\cos D+1$ is equal to :-

• A. ${\sin }^{2}A+{\cos }^{2}A$
• B. ${\sin }^{2}B+{\cos }^{2}B$
• C. ${\sin }^{2}A+{\sin }^{2}B$
• D. ${\cos }^{2}A+{\cos }^{2}B$

Answer 1

Answer: (A), (B)

The correct options are
A ${\sin }^{2}A+{\cos }^{2}A$
B ${\sin }^{2}B+{\cos }^{2}B$

We have,

$\begin{array}{c}A+C={180}^{0}B+D={180}^0\\ A=\pi -C\cos B=-\cos D\\ \cos A=-\cos C,\end{array}$

$\begin{array}{c}\cos A+\cos C+cosB+cosD=0\\ cosA+cosB+cosC+cosD+1=1\end{array}$

We have,

${\sin }^{2}A+{\cos }^{2}A=1$

Also ${\sin }^{2}B+{\cos }^{2}B=1$.