Question

If $\alpha ,\beta ,\gamma$ and $\delta$ be four angles of cyclic quadrilateral then the value of $\cos \alpha +\cos \beta +\cos \gamma +\cos \delta$ is

• A. $1$
• B. $0$
• C. $-1$
• D. none of these

Answer 1

The correct option is B $0$

Since,this is cyclic quadrilateral then

$\alpha +\gamma =\pi =\beta +\delta$ and $\alpha +\beta +\gamma +\delta =2\pi$ ...(i)

$\cos \alpha +\cos \beta +\cos \gamma +\cos \delta$

$=2\cos \frac{\alpha +\gamma }{2}\cos \frac{\alpha -\gamma }{2}+2\cos \frac{\beta +\delta }{2}\cos \frac{\beta -\delta }{2}$

From eq(i), we get,

$=2\cos \frac{\pi }{2}\cos \frac{\alpha -\gamma }{2}+2\cos \frac{\pi }{2}\cos \frac{\beta -\delta }{2}=0$

Hence, option 'B' is correct.