1. sin6x/cos8x=-1 2. sin12x/sin8x=-1

  [latex]1. frac{sin6x}{cos8x} = -1\\ cos8x e 0; 8x e frac{pi}{2} + pi n, n in Z; x e frac{pi}{16} + frac{pi n}{8}, n in Z\\ sin6x = -cos8x\\ sin6x + cos8x = 0\\ cos(frac{pi}{2} - 6x) + cos8x = 0\\ 2cos(frac{frac{pi}{2} - 6x + 8x}{2})cos(frac{frac{pi}{2} - 6x - 8x }{2}) = 0\\ cos(frac{pi}{4} + x)cos(frac{pi}{4} - 7x) = 0 [/latex]     [latex]1) cos(frac{pi}{4} + x) = 0\\ frac{pi}{4} + x = frac{pi}{2} + pi n, n in Z\\ x = frac{pi}{4} + pi n, n in Z \\ frac{pi}{4} + pi n e frac{pi}{16} + frac{pi k}{8}\\ n e frac{2k - 3}{16}\\ oxed{ x = frac{pi}{4} + pi n, n in Z setminus { frac{2k - 3}{16}| k in Z } } [/latex]     [latex]2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} e frac{pi}{16} + frac{pi k}{8}\\ n e -frac{14k + 11}{16}\\ oxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z setminus { -frac{14k + 11}{16}| k in Z } }[/latex]       [latex]2. frac{sin12x}{sin8x} = -1\\ sin8x e 0; 8x e pi n, n in Z; x e frac{pi n}{8}, n in Z\\ sin12x + sin8x = 0\\ 2sin(frac{12x + 8x}{2})cos(frac{12x-8x}{2}) = 0\\ sin(10x)cos(4x) = 0\\\ 1) sin(10x) = 0\\ 10x = pi n, n in Z,\\ frac{pi n}{10} e frac{pi k}{8}\\ n e frac{5k}{4} \\ oxed{ x = frac{pi n}{10}, n in Z setminus { frac{5k}{4} | k in Z } }[/latex]     [latex]2) cos(4x) = 0\\ 4x = frac{pi}{2} + pi n, n in Z\\ x = frac{pi}{8} + frac{pi n}{4}, n in Z\\ frac{pi}{8} + frac{pi n}{4} e frac{pi k}{8} \\ n e frac{k - 1}{2}\\ oxed{ x = frac{pi}{8} + frac{pi n}{4}, n in Z setminus {frac{k - 1}{2} | k in Z } } [/latex]