cos(8x)-cos(2x)=sin(5x)

Solution

cos (8 x) - cos (2 x) = sin (5 x)

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

Solution steps

cos (8 x) - cos (2 x) = sin (5 x)

Subtract

sin (5 x)

from both sides

cos (8 x) - cos (2 x) - sin (5 x) = 0

Rewrite using trig identities

- sin (5 x) - 2 sin (3 x) sin (5 x) = 0

Factor

- sin (5 x) - 2 sin (3 x) sin (5 x) : - sin (5 x) (2 sin (3 x) + 1)

- sin (5 x) (2 sin (3 x) + 1) = 0

Solving each part separately

sin (5 x) = 0 or 2 sin (3 x) + 1 = 0

sin (5 x) = 0 : x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}

2 sin (3 x) + 1 = 0 : x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{7 π}{18} + \frac{2 πn}{3}, x = \frac{11 π}{18} + \frac{2 πn}{3}

tan (x) = - \frac{1}{5}

2 + 2 cos (x) = 0

2 sin (θ) sec (θ) = 4 sin (θ)

sec (x - \frac{π}{7}) cos (x - \frac{π}{7}) = sin (x - \frac{π}{7})

cos (x) = \frac{17}{28}