If 4 cos² A - 3 = 0 and 0° ≤ A ≤ 90°; find:

(i) angle A

(ii) cos 3 A

(iii) tan² A + cos² A

(i) 4 cos² A - 3 = 0

⇒ 4 cos² A = 3

⇒ cos² A = $\dfrac{3}{4}$

⇒ cos A = $\sqrt{\dfrac{3}{4}}$

⇒ cos A = $\dfrac{\sqrt{3}}{2}$

⇒ cos A = cos 30°

⇒ A = 30°

Hence, the value of angle A = 30°.

(ii) cos 3A

= cos (3 x 30°)

= cos 90°

= 0

Hence, the value of cos 3A = 0.

(iii)

$= \text{tan}^2 \text{A} + \text{cos}^2 \text{A}\\[1em] = \text{tan}^2 30° + \text{cos}^2 30°\\[1em] = (\text{tan 30°})^2 + (\text{cos 30°})^2\\[1em] = \Big(\dfrac{1}{\sqrt{3}}\Big)^2 + \Big(\dfrac{\sqrt{3}}{2}\Big)^2\\[1em] = \dfrac{1}{3} + \dfrac{3}{4}\\[1em] = \dfrac{4 + 9}{12}\\[1em] = \dfrac{13}{12}\\[1em] = 1\dfrac{1}{12}$

Hence, the value of tan² A + cos² A = $1\dfrac{1}{12}$.