Question:
The general solution of the equation $7 \cos ^{2} x+3 \sin ^{2} x=4$ is
(a) $x=2 n \pi \pm \frac{\pi}{6}, n \in Z$
(b) $x=2 n \pi \pm \frac{2 \pi}{3}, n \in Z$
(c) $x=n \pi \pm \frac{\pi}{3}, n \in Z$
(d) none of these
Solution:
(c) $x=n \pi \pm \frac{\pi}{3}, n \in Z$
Given:
$7 \cos ^{2} x+3 \sin ^{2} x=4$
$\Rightarrow 7 \cos ^{2} x+3\left(1-\cos ^{2} x\right)=4$
$\Rightarrow 7 \cos ^{2} x+3-3 \cos ^{2} x=4$
$\Rightarrow 4 \cos ^{2} x+3=4$
$\Rightarrow 4\left(1-\cos ^{2} x\right)=3$
$\Rightarrow 4 \sin ^{2} x=3$
$\Rightarrow \sin ^{2} x=\frac{3}{4}$
$\Rightarrow \sin x=\frac{\sqrt{3}}{2}$
$\Rightarrow \sin x=\sin \frac{\pi}{3}$
$\Rightarrow x=n \pi \pm \frac{\pi}{3}, \mathrm{n} \in \mathrm{Z}$