General solution of equation $7{\cos }^2\theta +3{\sin }^2\theta =4$ is

$7{\cos }^2\theta +3{\sin }^2\theta =4 \Rightarrow 4{\cos }^2\theta +3{\cos }^2\theta +3{\sin }^2\theta =4$

$$\begin{gathered} \Rightarrow 4{\cos }^2\theta +3=4\Rightarrow 4{\cos }^2\theta =1\Rightarrow {\cos }^2\theta =\frac{1}{4}={\left(\frac{1}{2}\right)}^2 \\ \Rightarrow {\sin }^2\theta =\frac{3}{4} \Rightarrow \sin \theta =\pm \frac{\sqrt{3}}{2} \end{gathered}$$

$\Rightarrow \theta =n\pi \pm \frac{\pi }{3},n\in z$