If $$3 \cos^2 A + 7 \sin^2 A = 4$$, then what is the value of $$\cot A$$, given that A is an acute angle?

As per the given question,

$$3 \cos^2 A + 7 \sin^2 A = 4$$

$$\Rightarrow 3 \cos^2 A + 3 \sin^2 A +4 \sin^2 A = 4$$

We know that $$\sin^2 \theta+\cos^2\theta=1$$

$$\Rightarrow 3 (\cos^2 A +  \sin^2 A )+4 \sin^2 A = 4$$

$$\Rightarrow 3 (1 )+4 \sin^2 A = 4$$

$$\Rightarrow 4 \sin^2 A = 1$$

$$\Rightarrow \sin A = \dfrac{1}{2}=\sin\dfrac{\pi}{6}$$

$$\Rightarrow A=\dfrac{\pi}{6}$$

Hence, $$\cot\theta=\cot \dfrac{\pi}{6}=\sqrt3$$