If 4 sin² θ - 1 = 0 and angle θ is less than 90°, find the value of θ and hence the value of cos² θ + tan² θ.

4 sin² θ - 1 = 0

⇒ 4 sin² θ = 1

⇒ sin² θ = $\dfrac{1}{4}$

⇒ sin θ = $\sqrt{\dfrac{1}{4}}$

⇒ sin θ = $\dfrac{1}{2}$

⇒ sin θ = sin 30°

So, θ = 30°

Now, cos² θ + tan² θ

= cos² 30° + tan² 30°

$= \Big(\dfrac{\sqrt3}{2}\Big)^2 + \Big(\dfrac{1}{\sqrt3}\Big)^2\\[1em] = \dfrac{3}{4} + \dfrac{1}{3}\\[1em] = \dfrac{3 \times 3}{4 \times 3} + \dfrac{1 \times 4}{3 \times 4}\\[1em] = \dfrac{9}{12} + \dfrac{4}{12}\\[1em] = \dfrac{9 + 4}{12}\\[1em] = \dfrac{13}{12}\\[1em] = 1\dfrac{1}{12}$

Hence, θ = 30° and cos² 30° + tan² 30° = $\dfrac{13}{12}$ = $1\dfrac{1}{12}$.