Question

The minimum value of $({\sin }^2\theta +{\cos }^2\theta +{\sec }^2\theta +{\text{cosec}}^2\theta +{\tan }^2\theta +{\cot }^2\theta )$ is

Answer 1

$({\sin }^2\theta +{\cos }^2\theta +{\sec }^2\theta +{\text{cosec}}^2\theta +{\tan }^2\theta +{\cot }^2\theta )=(1+{\sec }^2\theta +{\text{cosec}}^2\theta +{\tan }^2\theta +{\cot }^2\theta )=(1+1+{\tan }^2\theta +1+{\cot }^2\theta +{\tan }^2\theta +{\cot }^2\theta )=(3+2{\tan }^2\theta +2{\cot }^2\theta )=3+2({\tan }^2\theta +{\cot }^2\theta )$
The value of above expression is minimum,when ${\tan }^2\theta +{\cot }^2\theta$ is minimum.
We know that ${\tan }^2\theta +{\cot }^2\theta \ge 2$
So, minimum value is $7.$