# If $p=\cos 55^{\large\circ},q=\cos 65^{\large\circ}$ and $r=\cos 175^{\large\circ}$ then the value of $\large\frac{1}{p}+\frac{1}{q}+\frac{r}{pq}$ is

$(a)\;0\qquad(b)\;-1\qquad(c)\;1\qquad(d)\;None\;of\;these$

Given that
$p=\cos 55^{\large\circ},q=65^{\large\circ},r=\cos 175^{\large\circ}$
Then $\large\frac{1}{p}+\frac{1}{q}+\frac{r}{pq}=\frac{P+q+r}{pq}$
$\qquad\qquad\qquad\quad\;\;\;=\large\frac{\cos 55^{\large\circ}+\cos 65^{\large\circ}+\cos 175^{\large\circ}}{\cos 55^{\large\circ}\cos 65^{\large\circ}}$
$\qquad\qquad\qquad\quad\;\;\;=\large\frac{\cos 55^{\large\circ}+2\cos\big(\Large\frac{175^{\large\circ}+65^{\large\circ}}{2}\big)\cos\big(\Large\frac{175^{\large\circ}-65^{\large\circ}}{2}\big)}{\cos 55^{\large\circ}\cos 65^{\large\circ}}$
$\qquad\qquad\qquad\quad\;\;\;=\large\frac{\cos 55^{\large\circ}+2\cos 120^{\large\circ}\cos 55^{\large\circ}}{\cos 55^{\large\circ}\cos 65^{\large\circ}}$
$\cos 120^{\large\circ}=-\large\frac{1}{2}$
$\qquad\qquad\qquad\quad\;\;\;=(1-2\times\large\frac{1}{2})$
$\qquad\qquad\qquad\quad\;\;\;=\large\frac{0}{\cos 65^{\large\circ}}$
$\qquad\qquad\qquad\quad\;\;\;=0$
Hence (a) is the correct answer.