Question

If $y=(\sin x+\text{cosec}x{)}^2+(\cos x+\sec x{)}^2$, then the minimum value of $y,\forall x\in R$, is

• A. 7
• B. 3
• C. 9
• D. 0

Answer 1

Answer: (C)

$y=\left({\sin }^{2}x+{\cos }^{2}x\right)+2(\sin x\text{cosec}x+\cos x\sec x)+{\sec }^{2}x+{\text{cosec}}^{2}x$
$=5+2+{\tan }^{2}x+{\cot }^{2}x$
$=7+(\tan x-\cot x{)}^2+2$
$\therefore y_{\min }=9$