In this question you must show all stages of your working - Edexcel - A-Level Maths Pure - Question 11 - 2020 - Paper 2

We start from the equation:

$1 − ext{cos } 3x = ext{sin } x$.

Substituting our earlier result for $ext{cos } 3x$ gives:

$1 − (4 ext{cos}^3 x - 3 ext{cos} x) = ext{sin } x$.

Rearranging this, we get:

$4 ext{cos}^3 x - 3 ext{cos} x + ext{sin } x - 1 = 0.$

Using the identity $$ ext{sin }^2 x + ext{cos}^2 x = 1$$$ we can express this entirely in terms of cosines by substituting for sin:

ext{sin } x = rac{1}{2}(1 - ext{cos } 2x).

This gives us a cubic equation in terms of $ext{cos } x$. To find the values of $x$ in the interval $−90^ ext{o} ≤ x ≤ 180^ ext{o}$, we can find the roots of this cubic equation, applying the necessary techniques (like factoring or synthetic division). The solutions to this equation must then be checked against the boundaries of the given interval.