A) 0
B) \[-1\]
C) 1
D) none of these
Correct Answer: A
Solution :
Given that \[p=\cos {{55}^{o}},q=\cos {{65}^{o}}\]and \[r=\cos {{175}^{o}}\] Then, \[\frac{1}{p}+\frac{1}{q}+\frac{r}{pq}=\frac{p+q+r}{pq}\] \[=\frac{\cos {{55}^{o}}+\cos {{65}^{o}}+\cos {{175}^{o}}}{\cos {{55}^{o}}\cos {{65}^{o}}}\] \[=\frac{\cos {{55}^{o}}+2\cos \left( \frac{{{175}^{o}}+{{65}^{o}}}{2} \right)\cos \left( \frac{{{175}^{o}}-{{65}^{o}}}{2} \right)}{\cos {{55}^{o}}\cos {{65}^{o}}}\] \[=\frac{\cos {{55}^{o}}+2\cos {{120}^{o}}\cos {{55}^{o}}}{\cos {{55}^{o}}\cos {{65}^{o}}}\] \[=\frac{\cos {{55}^{o}}(1+2cos{{120}^{o}})}{\cos {{55}^{o}}\cos {{65}^{o}}}\] \[=\frac{\left( 1-2\times \frac{1}{2} \right)}{\cos {{65}^{o}}}\] \[\left( \because \,\cos {{120}^{o}}=-\frac{1}{2} \right)\] \[=\frac{0}{\cos {{65}^{o}}}=0\]
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