Finally, combine like terms and set the equation equal to 0 to solve for the values of x:
6cos^2(x) - 6cos^3(x) - 5cosx + 1 = 0
However, this is a more complex trigonometric equation that may not have a simple algebraic solution. You may need to use numerical methods or a graphing calculator to find the values of x that satisfy this equation.
To solve this equation, you first need to distribute the terms:
1−cosx1 - cosx1−cosx4+3cos2x4 + 3cos2x4+3cos2x = 4 + 3cos2x - 4cosx - 3cosx*cos2x
Next, we can simplify the equation further by using the double angle identity cos2x = 2cos^2xxx - 1:
1−cosx1 - cosx1−cosx(4 + 3(2cos^2(x) - 1) - 4cosx - 3cosx(2cos^2(x) - 1)
Now, distribute and simplify the terms:
4 + 6cos^2(x) - 3 - 8cosx - 3cosx(2cos^2(x) - 1)
4 + 6cos^2(x) - 3 - 8cosx - 6cos^3(x) + 3cosx
Finally, combine like terms and set the equation equal to 0 to solve for the values of x:
6cos^2(x) - 6cos^3(x) - 5cosx + 1 = 0
However, this is a more complex trigonometric equation that may not have a simple algebraic solution. You may need to use numerical methods or a graphing calculator to find the values of x that satisfy this equation.