A) \[\theta +2\varphi =90{}^\circ \]
B) \[\theta +2\varphi =60{}^\circ \]
C) \[\theta +2\varphi =30{}^\circ \]
D) \[\theta +2\varphi =45{}^\circ \]
Correct Answer: D
Solution :
Given, \[\tan \theta =\frac{1}{7},\sin \varphi =\frac{1}{\sqrt{10}}\] \[\sin \theta =\frac{1}{\sqrt{50}},\,\,\cos \theta =\frac{7}{\sqrt{50}},\,\,\cos \varphi =\frac{3}{\sqrt{10}}\] \[\therefore \,\,\cos 2\varphi =2{{\cos }^{2}}\varphi -1=2.\frac{9}{10}-1=\frac{8}{10}\] \[\sin 2\varphi =2\sin \varphi \cos \varphi =2\times .\frac{1}{\sqrt{10}}\times \frac{3}{\sqrt{10}}=\frac{6}{10}\] \ \[\cos (\theta +2\varphi )=\cos \theta \cos 2\varphi -\sin \theta \sin 2\varphi \] \[=\frac{7}{\sqrt{50}}\times \frac{8}{10}-\frac{1}{\sqrt{50}}.\frac{6}{10}\] \[=\frac{56-6}{10\sqrt{50}}=\frac{50}{10\sqrt{50}}=\frac{5\sqrt{2}}{10}=\frac{1}{\sqrt{2}}\] \\[\theta +2\varphi ={{45}^{o}}\].
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