Question

$(\sin \theta +csc\theta {)}^2+(\cos \theta +sec\theta {)}^2$ is

• A. $\ge 9$
• B. $\le 9$
• C. $=9$
• D. $Noneofthese$

Answer 1

The correct option is A $\ge 9$

$(\sin \theta +csc\theta {)}^2+(\cos \theta +sec\theta {)}^2=({\sin }^2\theta +2+{\csc }^2\theta )+({\cos }^2\theta +2+{\sec }^2\theta )$

$=5+{\csc }^2\theta +{\sec }^2\theta =5+\frac{1}{{\sin }^2\theta }+\frac{1}{{\cos }^2\theta }=5+\frac{1}{{\cos }^2\theta {\sin }^2\theta }$

$=5+\frac{4}{4{\cos }^2\theta {\sin }^2\theta }=5+\frac{4}{{\sin }^{2}2\theta }=5+4{\csc }^{2}2\theta$

${\csc }^{2}2\theta \ge 1so,5+4{\csc }^{2}2\theta \ge 9$