A) \[\frac{7}{\sqrt{2}}\]
B) \[\frac{7}{5\sqrt{2}}\]
C) \[\frac{7}{\sqrt{5}}\]
D) \[\frac{7}{2\sqrt{5}}\]
Correct Answer: B
Solution :
We have \[\cos \theta =\frac{3}{5}\]and\[\cos \varphi =\frac{4}{5}\]. Therefore \[\cos (\theta -\varphi )=\cos \theta \cos \varphi +\sin \theta \sin \varphi \] \[=\frac{3}{5}.\frac{4}{5}+\frac{4}{5}.\frac{3}{5}=\frac{24}{25}\] But \[2{{\cos }^{2}}\left( \frac{\theta -\varphi }{2} \right)=1+\cos (\theta -\varphi )=1+\frac{24}{25}=\frac{49}{50}\] \\[{{\cos }^{2}}\left( \frac{\theta -\varphi }{2} \right)=\frac{49}{50}\]. Hence, \[\cos \left( \frac{\theta -\varphi }{2} \right)=\frac{7}{5\sqrt{2}}\].
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