question_answer

If \[\sin {{18}^{0}}=\frac{\sqrt{5}-1}{4},\] then what is the value of \[\sin 18{}^\circ \]?

A) \[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]

B) \[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\]

C) \[\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\]

D) \[\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}\]

Correct Answer: A

Solution :

\[\because \,\,\sin 18{}^\circ =\frac{\sqrt{5}-1}{4}\] \[{{x}^{2}}={{4}^{2}}-{{\left( \sqrt{5}-1 \right)}^{2}}\] \[\Rightarrow \,\,x=\sqrt{10+2\sqrt{5}}\] \[\Rightarrow \,\,\cos 18{}^\circ =\frac{\sqrt{10+2\sqrt{5}}}{4}\] \[\Rightarrow \,\,\,2\,{{\cos }^{2}}9-1=\frac{\sqrt{10+2\sqrt{5}}}{4}\] \[{{\cos }^{2}}9=\frac{\sqrt{10+2\sqrt{5}}+4}{8}\] \[\Rightarrow \,\,\,{{\sin }^{2}}81=\frac{4+\sqrt{10+2\sqrt{5}}}{8}\] After squaring all the options available, we come to a conclusion that option is correct.