Question

If $7{\sin }^{2}x+3{\cos }^{2}x=4$, show that $\tan x=\frac{1}{\sqrt{3}}$

Answer 1

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

$4{\sin }^{2}x+3{\sin }^{2}x+3{\cos }^{2}x=4$

$4{\sin }^{2}x+3=4$

$4{\sin }^{2}x=1$

${\sin }^{2}x=\frac{1}{4}$

$\sin x=\frac{1}{2}$ or $\sin x=-\frac{1}{2}$

Taking the positive root

$x=\frac{\pi }{6}$

$\tan \left(\frac{\pi }{6}\right)=\frac{1}{\sqrt{3}}$

Hence proved.