If 7 sin2 θ + 3 cos2θ = 4, show that tan θ

Given: $7 \sin ^{2} \theta+3 \cos ^{2} \theta=4$ and we have to prove that $\tan \theta=\frac{1}{\sqrt{3}}$

We can write the given expression as

$\left(7 \sin ^{2} \theta\right)+3 \cos ^{2} \theta=4$

$\Rightarrow \quad\left(4 \sin ^{2} \theta+3 \sin ^{2} \theta\right)+3 \cos ^{2} \theta=4$

$\Rightarrow \quad 4 \sin ^{2} \theta+3\left(\sin ^{2} \theta+\cos ^{2} \theta\right)=4$

$\Rightarrow \quad 4 \sin ^{2} \theta+3(1)=4$

$\Rightarrow \quad 4 \sin ^{2} \theta=4-3$

$\Rightarrow \quad \sin ^{2} \theta=\frac{1}{4}$

$\Rightarrow \quad \sin \theta=\pm \frac{1}{2}$

$\Rightarrow \quad 0=30^{\circ}$

Therefore,

$\tan 30^{\circ}=\frac{1}{\sqrt{3}}$

Hence proved