Question

Minimum value of the expression $(4{\sin }^2\theta +12{\text{cosec}}^2\theta -13)$ is equal to ​​​​​​

Answer 1

$(4{\sin }^2\theta +12{\text{cosec}}^2\theta -13)$
$=4[{\sin }^2\theta +{\text{cosec}}^2\theta ]+8{\text{cosec}}^2\theta -13$
$=4\left[\right(\sin \theta -\text{cosec}\theta {)}^2+2]+8{\text{cosec}}^2\theta -13$
$=4(\sin \theta -\text{cosec}\theta {)}^2+8{\text{cosec}}^2\theta -5$
Since, $(\sin \theta -\text{cosec}\theta {)}^2\ge 0,{\text{cosec}}^2\theta \ge 1$
Hence, minimum value will appear when $\sin \theta =\text{cosec}\theta \Rightarrow \sin \theta =\pm 1$
So, minimum value will be $8-5=3$