TRIGONOMETRIC IDENTITIES WITHOUT VARIABLES

 

               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) $sin \frac{\pi}{14}\cdot sin\frac{3\pi}{14} \cdot sin \frac{5\pi}{14}=\frac{1}{8}.$

               4.

$$1^\circ .\;\;cos\frac{\pi}{21}-cos \frac{4\pi}{21}+cos \frac{5\pi}{21}=\frac{\sqrt{21}-1}{4};$$

$$1^{\circ \circ}.\;\;cos \frac{2\pi}{21}+cos\frac{8\pi}{21}+cos\frac{10\pi}{21}=\frac{\sqrt{21}+1}{4};$$

$$2^\circ\;\;-sin\frac{\pi}{21}+sin\frac{4\pi}{21}+sin\frac{5\pi}{21}=\frac{1}{2} \sqrt{\frac{5+\sqrt{21}}{2}};$$

$$3^\circ.\;\;sin\frac{2\pi}{21}+sin\frac{8\pi}{21}-sin\frac{10\pi}{21}=\frac{1}{2}\sqrt{\frac{5-\sqrt{21}}{2}};$$

$$4^\circ\;\;cos\frac{\pi}{21}\cdot cos\frac{4\pi}{21}\cdot cos\frac{5\pi}{21}=\frac{5+\sqrt{21}}{16};$$

$$5^\circ.\;\;cos\frac{2\pi}{21}\cdot cos\frac{8\pi}{21}\cdot cos\frac{10\pi}{21}=\frac{5-\sqrt{21}}{16};$$

$$6^\circ.\;\;sin\frac{\pi}{21}\cdot sin\frac{4\pi}{21}\cdot sin\frac{5\pi}{21}=\frac{1}{8}\sqrt{\frac{5-\sqrt{21}}{2}}.$$

          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)       $\sin \frac{\pi}{2n+1} \cdot \sin \frac{2\pi}{2n+1} \dots \sin \frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$