Solve for x :

cos² 30° + sin² 2x = 1

cos² 30° + sin² 2x = 1

$⇒ \Big(\dfrac{\sqrt3}{2}\Big)^2 + \text{sin }^2 2x = 1\\[1em] ⇒ \Big(\dfrac{3}{4}\Big) + \text{sin }^2 2x = 1\\[1em] ⇒ \text{sin}^2 2x = 1 - \dfrac{3}{4}\\[1em] ⇒ \text{sin}^2 2x = \dfrac{4}{4} - \dfrac{3}{4}\\[1em] ⇒ \text{sin}^2 2x = \dfrac{4 - 3}{4}\\[1em] ⇒ \text{sin}^2 2x = \dfrac{1}{4}\\[1em] ⇒ \text{sin} 2x = \sqrt\dfrac{1}{4}\\[1em] ⇒ \text{sin} 2x = \dfrac{1}{2}\\[1em] ⇒ \text{sin} 2x = \text{sin }30°$

So, 2x = 30°

⇒ x = $\dfrac{30°}{2}$

⇒ x = 15°

Hence, x = 15°.