1.(Euclid)
$$sin^2 18^\circ+sin^2 30^\circ=sin^2 36^\circ$$
or, expressing angles in radians
$$sin^2 \frac{\pi}{10}+sin^2 \frac{\pi}{6}=sin^2 \frac{\pi}{5}.$$
2.
$$cos 36^\circ -cos 72^\circ =\frac{1}{2}$$
or, expressing angles in radians
$$cos \frac{\pi}{5}-cos\frac{2\pi}{5}=\frac{1}{2}.$$
3.
i) $cos \frac{\pi}{7} -cos \frac{2\pi}{7}+cos \frac{3\pi}{7}=\frac{1}{2};$
ii) $sin \frac{\pi}{14}-sin \frac{3\pi}{14}+sin \frac{5\pi}{14}=\frac{1}{2};$
iii) $cos \frac{\pi}{7} \cdot cos \frac{2\pi}{7} \cdot cos \frac{3\pi}{7}=\frac{1}{8};$
iv) $sin \frac{\pi}{14}\cdot sin\frac{3\pi}{14} \cdot sin \frac{5\pi}{14}=\frac{1}{8}.$
$$1^\circ .\;\;cos\frac{\pi}{21}-cos \frac{4\pi}{21}+cos \frac{5\pi}{21}=\frac{\sqrt{21}-1}{4};$$
$$1^{\circ \circ}.\;\;cos \frac{2\pi}{21}+cos\frac{8\pi}{21}+cos\frac{10\pi}{21}=\frac{\sqrt{21}+1}{4};$$
$$2^\circ\;\;-sin\frac{\pi}{21}+sin\frac{4\pi}{21}+sin\frac{5\pi}{21}=\frac{1}{2} \sqrt{\frac{5+\sqrt{21}}{2}};$$
$$3^\circ.\;\;sin\frac{2\pi}{21}+sin\frac{8\pi}{21}-sin\frac{10\pi}{21}=\frac{1}{2}\sqrt{\frac{5-\sqrt{21}}{2}};$$
$$4^\circ\;\;cos\frac{\pi}{21}\cdot cos\frac{4\pi}{21}\cdot cos\frac{5\pi}{21}=\frac{5+\sqrt{21}}{16};$$
$$5^\circ.\;\;cos\frac{2\pi}{21}\cdot cos\frac{8\pi}{21}\cdot cos\frac{10\pi}{21}=\frac{5-\sqrt{21}}{16};$$
$$6^\circ.\;\;sin\frac{\pi}{21}\cdot sin\frac{4\pi}{21}\cdot sin\frac{5\pi}{21}=\frac{1}{8}\sqrt{\frac{5-\sqrt{21}}{2}}.$$
5. a) $\sin \frac{\pi}{n} \cdot \sin \frac{2\pi}{n} \dots \sin \frac{(n-1)\pi}{n}=\frac{n}{2^{n-1}}$
b) $\sin \frac{\pi}{2n} \cdot \sin \frac{2\pi}{2n} \dots \sin \frac{(n-1)\pi}{2n}= \frac{\sqrt{n}}{2^{n-1}}$
c) $\sin \frac{\pi}{2n+1} \cdot \sin \frac{2\pi}{2n+1} \dots \sin \frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$
6. $\tan10^{\circ}\cdot \tan50^{\circ}\cdot \cot20^{\circ}=\frac{1}{\sqrt{3}}\;\;;\;\;\frac{\tan10^{\circ}}{\tan20^{\circ}\cdot \tan40^{\circ}}=\frac{1}{\sqrt{3}}$ (!!)
7. a) $\tan20^{\circ}-\tan40^{\circ}+\tan80^{\circ}=3\sqrt{3};$
b) $\tan20^{\circ}\cdot \tan40^{\circ}+\tan40^{\circ}\cdot \tan80^{\circ}-\tan20^{\circ}\cdot \tan80^{\circ}=3;$
c) $\tan20^{\circ}\cdot \tan40^{\circ}\cdot \tan80^{\circ}=\sqrt{3}.$
8. $\frac{\sin 110^{\circ}}{\sin 30^{\circ}}-\frac{\sin 60^{\circ}}{\sin 80^{\circ}}=1.$
9. a) $\cos \frac{\pi}{n}\cdot \cos \frac {3\pi}{n}\dots \cos \frac{(2n-1)\pi}{n}=\frac{1}{2^{n-1}}\left [\cos \frac{n\pi}{2}+(-1)^n\ \right ] $
b) $\cos \frac{2\pi}{n}\cdot \cos \frac{4\pi}{n}\dots \cos \frac{2(n-1)\pi}{n}=\frac{1}{2^{n-1}}\left [\cos \frac{n\pi}{2}-(-1)^n \right ]$
c)
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