A) \[cot\frac{\theta }{2}\]
B) \[\sec \frac{\theta }{2}\]
C) \[\sin \frac{\theta }{2}\]
D) \[\cos \frac{\theta }{2}\]
Correct Answer: D
Solution :
(d): \[\frac{1}{2}\sqrt{1+\cos \theta }+\frac{1}{2}\sqrt{1-\cos \theta }\] \[=\frac{1}{2}\left( 1+cos{{60}^{{}^\circ }}+1-cos{{60}^{{}^\circ }} \right)\] \[=\frac{1}{2}\left( \sqrt{1+\frac{\sqrt{3}}{2}}+\sqrt{1-\frac{\sqrt{3}}{2}} \right)\] \[=\frac{1}{2\sqrt{2}}\left( \sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}} \right)\] \[=\frac{1}{2\sqrt{2}}\times \frac{1}{2}\left( \sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}} \right)\] \[=\frac{1}{4}\sqrt{{{\left( \sqrt{3}+1 \right)}^{2}}}+\sqrt{{{\left( \sqrt{3}-1 \right)}^{2}}}\] \[=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}=\cos {{30}^{{}^\circ }}\] \[=\cos \frac{\theta }{2}\]
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