Question

Let $f\left(x\right)={\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}$ and $g\left(x\right)={\sec }^{2}x-{\tan }^{2}x.$ The two functions are equal over the set

• A. $\Phi$
• B. $R$
• C. $R-\{x|x=(2n+1)\frac{\pi }{2},n\in Z\}$
• D. None of these

Answer 1

Answer: (C)

$R-\{x|x=(2n+1)\frac{\pi }{2},n\in Z\}$

We know,
$f\left(x\right)=si{n}^2\frac{x}{2}+co{s}^2\frac{x}{2}=1$ for all values of $x$ and $g\left(x\right)=se{c}^{2}x-ta{n}^{2}x=1$ except $x=(2n+1)\frac{\pi }{2}wherenϵZ$

Therefore, two functions are equal in the interval:

$R-\{x|x=(2n+1)\frac{\pi }{2},nϵZ\}$

Hence, option 'C' is correct.