Question

(sin(7pi/24)cos(pi/24)-cos(7pi/24)sin(pi/24))/(cos(pi/7)cos(4pi/21)-sin(pi/7)sin(4pi/21))

Answer 1

$\frac{\sin \left(\frac{7\pi }{24}\right)\cos \left(\frac{\pi }{24}\right)-\cos \left(\frac{7\pi }{24}\right)\sin \left(\frac{\pi }{24}\right)}{\cos \left(\frac{\pi }{7}\right)\cos \left(\frac{4\pi }{21}\right)-\sin \left(\frac{\pi }{7}\right)\sin \left(\frac{4\pi }{21}\right)}$

$=\frac{\sin \left(\frac{7\pi }{24}\right)\cos \left(\frac{\pi }{24}\right)-\cos \left(\frac{7\pi }{24}\right)\sin \left(\frac{\pi }{24}\right)}{\cos \left(\frac{\pi }{7}+\frac{4\pi }{21}\right)}$

$=\frac{\cos \left(\frac{\pi }{24}\right)\sin \left(\frac{7\pi }{24}\right)-\sin \left(\frac{\pi }{24}\right)\cos \left(\frac{7\pi }{24}\right)}{\cos \left(\frac{\pi }{3}\right)}$

$=\frac{\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2}\cdot \frac{\sqrt{2}\sqrt{4-\sqrt{2}+\sqrt{6}}}{4}-\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}{2}\cdot \frac{\sqrt{2}\sqrt{4+\sqrt{2}-\sqrt{6}}}{4}}{\frac{1}{2}}$

$=\frac{\sqrt{2}\left(\sqrt{\sqrt{2+\sqrt{3}}+2}\sqrt{4+\sqrt{6}-\sqrt{2}}-\sqrt{-\sqrt{2+\sqrt{3}}+2}\sqrt{4+\sqrt{2}-\sqrt{6}}\right)}{4}$