Prove that ; `4 sin 27^@=(5+sqrt(5))-sqrt((3-sqrt(5)))` we have

Explanation

lets take
`=(sin 27^{circ}+cos 27^{circ})^{2}=sin ^{2} 27^{circ}+cos ^{2} 27^{circ}+2 sin 27^{circ} cos 27^{circ}=1+sin 54^{circ} (using identity sin 2 theta=2 sin theta cos theta )`



`=arrow(sin 27^{circ}+cos 27^{circ})^{2}=1+sin 54^{circ}=1+frac{sqrt{5}+1}{4}=frac{1}{4}(5+sqrt{5})`




`=arrow sin 27^{circ}+cos 27^{circ}=frac{1}{2} sqrt{5+sqrt{5}} `{...(1) }



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