Question

If $f\left(x\right)={\cos }^{2}x+{\sec }^{2}x$, then

• A. $f\left(x\right)<1$
• B. $f\left(x\right)=1$
• C. $1<f\left(x\right)<1$
• D. $f\left(x\right)\ge 2$

Answer 1

Answer: (D)

$f\left(x\right)\ge 2$

$f\left(x\right)={\cos }^{2}x+{\sec }^{2}x$

$f\left(x\right)=1-{\sin }^{2}x+{\tan }^{2}x+1$

$\Rightarrow 2+\frac{{\sin }^{2}x}{{\cos }^{2}x}=-{\sin }^{2}x$

$\Rightarrow 2+\frac{{\sin }^{2}x-{\sin }^{2}x{\cos }^{2}x}{{\cos }^{2}x}$

$\Rightarrow 2+\frac{{\sin }^{2}x(1-{\cos }^{2}x)}{{\cos }^{2}x}$

$\Rightarrow 2+\frac{{\sin }^{4}x}{{\cos }^{2}x}$

$\frac{{\sin }^{4}x}{{\cos }^{2}x}$

$\therefore f\left(x\right)\ge 2$