If $f\left(\theta \right)={\left(\sin \theta +\cos ec\theta \right)}^2+{\left(\cos \theta +\sec \theta \right)}^2$ , then minimum value of $f\left(\theta \right)$ is k, then k is divisible by

$f\left(\theta \right)={\sin }^2\theta +\cos e{c}^2\theta +2+{\cos }^2\theta +{\sec }^2\theta +2$

$=5+1+{\cot }^2\theta +1+{\tan }^2\theta$

$=7+{\tan }^2\theta +{\cot }^2\theta$

We know $\frac{{\tan }^2\theta +{\cot }^2\theta }{2}\ge \sqrt{{\tan }^2\theta {\cot }^2\theta }$

${\tan }^2\theta +{\cot }^2\theta \ge 2$

$\therefore$ Minimum value of $f\left(0\right)=7+2=9$

$\therefore$  $k=9$  is divisible by 1,3,9