Question

G = cos\(\dfrac{2\pi}{31}\) . cos\(\dfrac{4\pi}{31}\) . cos\(\dfrac{8\pi}{31}\) . cos\(\dfrac{16\pi}{31}\) . cos\(\dfrac{32\pi}{31}\)

Answer 1

\(G=cos\left(\dfrac{2\Omega}{31}\right)\cdot cos\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)\)

\(=\dfrac{2\cdot sin\left(\dfrac{2\Omega}{31}\right)\cdot cos\left(\dfrac{2\Omega}{31}\right)\cdot cos\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{2\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{sin\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{2\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{2\cdot sin\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{4\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{4\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{sin\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{4\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{2\cdot sin\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{8\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{8\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{sin\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{8\cdot sin\left(\dfrac{2\Omega}{31}\right)}=\dfrac{2\cdot sin\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{16\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{16\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{sin\left(\dfrac{32\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{16\cdot sin\left(\dfrac{2\Omega}{31}\right)}=\dfrac{2\cdot sin\left(\dfrac{32\Omega}{31}\right)\cdot cos\left(\dfrac{32\Omega}{31}\right)}{32\cdot sin\left(\dfrac{2\Omega}{31}\right)}=\dfrac{sin\left(\dfrac{64\Omega}{31}\right)}{32\cdot sin\left(\dfrac{2\Omega}{31}\right)}\)

\(=\dfrac{1}{2}\)