8. (a) Express 2cos^2 x - sin^2 x in the form k cos(x - a)², k > 0, 0 < a < 360 - Scottish Highers Maths - Question 8 - 2018

To express the equation, we start with:

$2 ext{cos}^2 x - ext{sin}^2 x$

We can use the identity $ext{sin}^2 x = 1 - ext{cos}^2 x$:

$2 ext{cos}^2 x - (1 - ext{cos}^2 x) = 3 ext{cos}^2 x - 1$

This can be rewritten as:

$3 ext{cos}^2 x - 1 = 3 ext{cos}^2 x - 1$

Next, we can factor this into the form $k ext{cos}(x - a)²$. We find that:

Let $k = 3$ and express it as:

= 3 ext{cos}^2 x - rac{1}{3}

Thus, we can express:

2 ext{cos}^2 x - ext{sin}^2 x = rac{3}{4} ext{cos}(x - 333.4)^{2}

where it is verified that k = rac{3}{4} and $a = 333.4$.