Solve for x :

cos² 30° + cos² x = 1

cos² 30° + cos² x = 1

$⇒ \Big(\dfrac{1}{2}\Big)^2 + \text{cos }^2 x = 1\\[1em] ⇒ \Big(\dfrac{1}{4}\Big) + \text{cos }^2 x = 1\\[1em] ⇒ \text{cos }^2 x = 1 - \Big(\dfrac{1}{4}\Big)\\[1em] ⇒ \text{cos }^2 x = \dfrac{4}{4} - \dfrac{1}{4}\\[1em] ⇒ \text{cos }^2 x = \dfrac{4 - 1}{4}\\[1em] ⇒ \text{cos } x = \sqrt\dfrac{3}{4}\\[1em] ⇒ \text{cos } x = \dfrac{\sqrt3}{2}\\[1em] ⇒ \text{cos } x = \text{cos 60°} \\[1em]$

So, x = 60°

Hence, x = 60°.