Maximum value of the expression ${\log }_3\left(9-2{\cos }^2\theta -4{\sec }^2\theta \right)$  is equal to________

For the expression  $a{\cos }^2\theta +b{\sec }^2\theta$  if  $b>a,$   then minimum value attains at  ${\cos }^2\theta ={\sec }^2\theta =1$

$\Rightarrow \max of\left\{9-\left(2{\cos }^2\theta +4{\sec }^2\theta \right)\right\}=3$

So, maximum of  ${\log }_3\left(9-2{\cos }^2\theta -4{\sec }^2\theta \right)={\log }_{3}3=1$