A) \[\frac{1}{2}\]
B) \[\frac{1}{\sqrt{3}}\]
C) \[\sqrt{3}\]
D) \[0\]
E) \[1\]
Correct Answer: D
Solution :
\[cos\text{ }20{}^\circ +cos\text{ }100{}^\circ +cos\text{ }140{}^\circ \] \[\Rightarrow \] \[cos\text{ }20{}^\circ +cos(90{}^\circ +10{}^\circ )+cos(90{}^\circ +50{}^\circ )\] \[=cos\text{ }20{}^\circ -sin\text{ }10{}^\circ -sin\text{ }50{}^\circ \] \[=cos\text{ }20{}^\circ -(sin\text{ }10{}^\circ +sin\text{ }50{}^\circ )\] \[=cos\text{ }20{}^\circ -2\text{ }sin\text{ }30{}^\circ \,cos\text{ }20{}^\circ \] \[(\because \sin C+\sin D=2\sin \frac{(C+D)}{2}.\cos \frac{(C-D)}{2})\] \[=\cos 20{}^\circ -2.\frac{1}{2}\cos 20{}^\circ \] \[=\cos 20{}^\circ -\cos 20{}^\circ =0\]
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