A) \[\frac{1}{4}\tan 10{}^\circ \]
B) \[\frac{1}{8}\cot 10{}^\circ \]
C) \[\frac{1}{8}\text{cosec}10{}^\circ \]
D) \[\frac{1}{8}\sec 10{}^\circ \]
Correct Answer: B
Solution :
\[x=\cos \,\,{{10}^{o}}\,\cos \,\,{{20}^{o}}\,\,\cos \,\,{{40}^{o}}\] \[=\frac{1}{2\,\,\sin \,\,{{10}^{o}}}\,[2\,\,\sin \,\,{{10}^{o}}\cos \,\,{{10}^{o}}\cos \,\,{{20}^{o}}\,\,\cos \,\,{{40}^{o}}]\] \[=\frac{1}{2\,.\,2\,\,\sin \,\,{{10}^{o}}}\,[2\,\,\sin \,\,{{20}^{o}}\cos \,\,{{20}^{o}}\,\,\cos \,\,{{40}^{o}}]\] \[=\frac{1}{2\,.\,4\sin {{10}^{o}}}[2\sin {{40}^{o}}\cos {{40}^{o}})=\frac{1}{8\sin {{10}^{o}}}(\sin {{80}^{o}})\] \[=\frac{1}{8\,\,\sin \,\,{{10}^{o}}}\cos \,\,{{10}^{o}}=\frac{1}{8}\cot \,\,{{10}^{o}}\].
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