How do you find the possible values of cos(x) from 5cos^2(x) - cos(x) = sin^2(x)?

First, you start by replacing sin^2(x) with 1-cos^2(x) as you want the equation to be in terms of cos(x) and you know sin^2(x)+cos^2(x) = 1. Then you rearrange the equation to get 0 on one side so that you can solve it. This gives 6cos^2(x) - cos(x) - 1 = 0. Then you can factorise this equation to give (2cos(x)-1)(3cos(x) + 1) = 0. Therefore, you'd get solutions when either 2cos(x) - 1 = 0 or when 3cos(x) + 1 = 0. You can rearrange these to find that cos(x) = 1/2 or -1/3.