A) \[\frac{23}{17}\left( \frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}} \right)\]
B) \[\frac{23}{17}\left( \frac{\sqrt{3}+1}{2}+\frac{1}{\sqrt{2}} \right)\]
C) \[\frac{23}{17}\left( \frac{\sqrt{3}-1}{2}-\frac{1}{\sqrt{2}} \right)\]
D) \[\frac{23}{17}\left( \frac{\sqrt{3}+1}{2}-\frac{1}{\sqrt{2}} \right)\]
Correct Answer: A
Solution :
Since \[\cos \theta =\frac{8}{17}\] and \[0<\theta <\frac{\pi }{2}\] \[\Rightarrow \,\,\sin \theta =\sqrt{1-\frac{{{8}^{2}}}{{{17}^{2}}}}=\frac{15}{17}\] The value of the given expression \[=\cos \,\,{{30}^{o}}\,.\,\cos \theta -\sin \,\,{{30}^{o}}\sin \theta +\cos \,\,{{45}^{o}}\cos \theta \] \[+\sin \,\,{{45}^{o}}\sin \theta +\cos \,\,{{120}^{o}}\cos \theta +\sin \,\,{{120}^{o}}\sin \theta \] \[=\cos \theta \,\left( \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}-\frac{1}{2} \right)-\sin \theta \,\left( \frac{1}{2}-\frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2} \right)\] \[=\frac{8}{17}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}-\frac{1}{2} \right)+\frac{15}{17}\,\left( \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}-\frac{1}{2} \right)\] \[=\frac{23}{17}\,\left( \frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}} \right)\].
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