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Question Number 110545 by peter frank last updated on 29/Aug/20

Commented by Dwaipayan Shikari last updated on 29/Aug/20

$$\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{3}}}$$

Commented by Dwaipayan Shikari last updated on 29/Aug/20

$$\frac{e^{\frac{\pi i}{24}}-e^{\frac{-\pi i}{24}}}{2i}$$

Answered by Dwaipayan Shikari last updated on 29/Aug/20

$$1+cos\frac{\pi }{6}=2{cos}^2\frac{\pi }{12}$$$$(1+\frac{\sqrt{3}}{2})=2{cos}^2\frac{\pi }{12}$$$$\sqrt{\frac{1}{2}(1+\frac{\sqrt{3}}{2})}=cos\frac{\pi }{12}=\frac{1}{2}\sqrt{2+\sqrt{3}}$$$$1-cos\frac{\pi }{12}=2{sin}^2\frac{\pi }{24}$$$$1-\frac{1}{2}\sqrt{2+\sqrt{3}}=2{sin}^2\frac{\pi }{24}$$$$\sqrt{\frac{1}{2}(1-\frac{1}{2}\sqrt{2+\sqrt{3}})}=sin\frac{\pi }{24}$$$$\frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{3}}}=sin\frac{\pi }{24}$$

Commented by peter frank last updated on 29/Aug/20

$$\mathrm{thank}\mathrm{you}$$

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