What is the simplified value of $\frac{7}{\sec ^{2} \theta}+\frac{3}{1+\cot ^{2} \theta}+4 \sin ^{2} \theta ?$

Expression $=\frac{7}{\sec ^{2} \theta}+\frac{3}{1+\cot ^{2} \theta}+4 \sin ^{2} \theta$

$=7 \cos ^{2} \theta+\frac{3}{\operatorname{cosec}^{2} \theta}+4 \sin ^{2} \theta$

$\left[\because \operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\right]$

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

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

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