Question

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

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

Answer 1

Answer: (A)

Given that
$p=\cos 55^{\circ },q=\cos 65^{\circ }$ and $r=\cos 175^{\circ }$
Then, $\frac{1}{p}+\frac{1}{q}+\frac{r}{pq}=\frac{p+q+r}{pq}$
$=\frac{\cos 55^{\circ }+\cos 65^{\circ }+\cos 175^{\circ }}{\cos 55^{\circ }\cos 65^{\circ }}$
$=\frac{\cos 55^{\circ }+2\cos \left(\frac{175^{\circ }+65^{\circ }}{2}\right)\cos \left(\frac{175^{\circ }-65^{\circ }}{2}\right)}{\cos 55^{\circ }\cos 65^{\circ }}$
$=\frac{\cos 55^{\circ }+2\cos 120^{\circ }\cos 55^{\circ }}{\cos 55^{\circ }\cos 65^{\circ }}$
$=\frac{\cos 55^{\circ }\left(1+2\cos 120^{\circ }\right)}{\cos 55^{\circ }\cos 65^{\circ }}$
$=\frac{\left(1-2\times \frac{1}{2}\right)}{\cos 65^{\circ }}\left(\because \cos 120^{\circ }=-\frac{1}{2}\right)$
$=\frac{0}{\cos 65^{\circ }}=0$