cos(4theta)-cos(6theta)=0

Question

cos(4θ)-cos(6θ) on the interval 0≤θ<2π

Answer 1

We know that cosine is an even function, so therefore:

 

cos 6θ = cos 4θ

 

Case 1: 6θ = 2kπ + 4θ

 

2θ = 2kπ

 

θ = kπ

 

Case 2: 6θ = 2kπ - 4θ

 

10θ = 2kπ

θ = kπ/5

 

Since the solution set from case 1 is a subset of the one of case 2, the general solution is just:

 

θ ∈ {kπ/5 | k ∈ Z}.

 

Finally, we need to list the answers in 0≤θ<2π:

 

θ ∈ {0,π/5,2π/5,3π/5,4π/5,π,6π/5,7π/5,8π/5,9π/5}.