\(\sin54^{o}=\sin(90^{o}-36^{o})=\cos36^{o}\)
=\(\frac{\sqrt{5}+1}{4}\)
Thus \(\sin54^{o}=\frac{\sqrt{5}+1}{4}\)
\(\cos54^{o}=\sqrt{1-\sin^{2}54^{o}}\)
=\(\sqrt{1-\left(\frac{\sqrt{5}+1}{4}\right)^{2}}\)
=\(\frac{1}{4}\sqrt{16-5-2\sqrt{5}-1}\)
=\(\frac{1}{4}\sqrt{10-2\sqrt{5}}\)
\(\tan54^{o}\)=\(\frac{\sin54^{o}}{\cos54^{o}}\)
=\(\frac{\sqrt{5}+1}{\sqrt{10-2\sqrt{5}}}\)
\([\) \(\because\) \(\sin54^{o}=\frac{\sqrt{5}+1}{4}\) and \(\cos54^{o}=\frac{1}{4}\sqrt{10-2\sqrt{5}}\)\(]\)
\(\sin72^{o}=\sin\left(90^{o}-18^{o}\right)\)
=\(\cos18^{o}=\frac{1}{4}\sqrt{10+2\sqrt{5}}\)
\(\cos72^{o}=\cos\left(90^{o}-18^{o}\right)\)
=\(\sin18^{o}=\frac{\sqrt{5}-1}{4}\)
\(\tan72^{o}=\frac{\sin72^{o}}{\cos72^{o}}\)
=\(\frac{\sqrt{10+2\sqrt{5}}}{\sqrt{5}-1}\)
\( [\) \(\because\) \(\sin72^{o}=\frac{1}{4}\sqrt{10+2\sqrt{5}}\) and
\(\cos72^{o}=\frac{\sqrt{5}-1}{4}\)\(]\)