Determine the exact value of $\cos(x−\pi)+\cos\left(x−\frac \pi2\right)$

Question

Consider $x \in(\pi; 2\pi)$ such that $\;3\cos\left(\dfrac x2\right)=-\sqrt2$

Determine the exact value of $\cos(x-\pi)+\cos\left(x-\dfrac\pi2\right)$.

I can't seem to find a way of solving this, Ii have tried using the angle summation formulas, etc.

Answer 1

By the duplication formula,

$$\cos x=2\left(-\frac{\sqrt2}{3}\right)^2-1=-\frac59=-\cos(x-\pi)$$ and $$\sin x=-2\frac{\sqrt2}{3}\sqrt{1-\left(-\frac{\sqrt2}{3}\right)^2}=-\frac{2\sqrt{14}}9=\cos(x-\frac\pi2).$$

(We take the positive sign for the sine because $x$ is known to exceed $\pi$.)

Answer 2

HINT:

Compound Angle Formulae

$$\cos(A\pm B) \equiv \cos A \cos B \mp \sin A \sin B$$