sin810=cos902cos90=1+sin180+1-sin1802cos90=1+5-14+1-5-142cos90=3+52+5-52cos90=3+5+5-54=cos810

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$\sin 81^{0}$

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$\frac{\sqrt{3 - \sqrt{5}} + \sqrt{5 + \sqrt{5}}}{4}$

$\frac{\sqrt{3 + \sqrt{5}} + \sqrt{5 - \sqrt{5}}}{4}$

$\frac{\sqrt{3 - \sqrt{5}} - \sqrt{5 + \sqrt{5}}}{4}$

$\frac{\sqrt{3 + \sqrt{5}} - \sqrt{5 - \sqrt{5}}}{4}$

answer is A.

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Detailed Solution

$\begin{array}{rcl} \sin 81^{0} & = & \cos 9^{0} \\ 2 \cos 9^{0} & = & \sqrt{1 + \sin 18^{0}} + \sqrt{1 - \sin 18^{0}} \\ 2 \cos 9^{0} & = & \sqrt{1 + \frac{\sqrt{5} - 1}{4}} + \sqrt{1 - \frac{\sqrt{5} - 1}{4}} \\ 2 \cos 9^{0} & = & \frac{\sqrt{3 + \sqrt{5}}}{2} + \frac{\sqrt{5 - \sqrt{5}}}{2} \\ \cos 9^{0} & = & \frac{\sqrt{3 + \sqrt{5}} + \sqrt{5 - \sqrt{5}}}{4} = \cos 81^{0} \end{array}$