A) 0
B) 1
C) \[\sqrt{2}\]
D) \[\sqrt{3}\]
Correct Answer: D
Solution :
\[\sin {{20}^{o}}(4+\sec {{20}^{o}})\] \[=\sin {{20}^{o}}\left( 4+\frac{1}{\cos {{20}^{o}}} \right)\] \[=\sin {{20}^{o}}\left( \frac{4\cos {{20}^{o}}+1}{\cos {{20}^{o}}} \right)\] \[=\frac{2\sin {{40}^{o}}+\sin {{20}^{o}}}{\cos {{20}^{o}}}\] \[=\frac{\sin {{40}^{o}}+\sin {{40}^{o}}+\sin {{20}^{o}}}{\cos {{20}^{o}}}\] \[\frac{\sin {{40}^{o}}+2\sin {{30}^{o}}\cos {{10}^{o}}}{\cos {{20}^{o}}}\] \[=\frac{\sin {{40}^{o}}+\cos {{10}^{o}}}{\cos {{20}^{o}}}\] \[=\frac{\sin {{50}^{o}}+\cos {{10}^{o}}}{\cos {{20}^{o}}}\] \[=\frac{2\cos {{30}^{o}}\cos {{20}^{o}}}{\cos {{20}^{o}}}\] \[=2\times \cos {{30}^{o}}\] \[=2\times \frac{\sqrt{3}}{2}=\sqrt{3}\]
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