Find all solutions to cos (5α) − cos (α) = sin (3α) on 0<α<π

Answer 1

cos(5α) - cos(α) =

cos(3α + 2α) - cos(3α - 2α) =

cos(3α) cos(2α) - sin(3α) sin(2α) -

(cos(3α) cos(2α) + sin(3α) sin(2α)) =

-2sin(3α) sin(2α)

Then, if -2sin(3α) sin(2α) = sin(3α),

sin(3α)(1 +2 sin(2α)) = 0

As sin(α) = 0 when α = n π,

sin(3α) = 0 when α = nπ/3, n an integer.

If (1 +2 sin(2α)) = 0,

sin(2α)) = -1./2

As sin(α) = -1/2 when α = 7π/6 or 11π/6

sin(2α) = -1/2 when α = 7π/12 or 11π/12 (In general, the solutions will be 7π/12 + nπ, 11π/12 + nπ, but these are the solutions in(0, π)

Thus, the solutions are π/3, 2π/3, 7π/6, 11π/6