Question

The value of $f\left(x\right)=\left[\frac{{\sin }^{2}x}{2}\right]+\left[\frac{{\cos }^{2}x}{2}\right]$, where $[.]$ denotes greatest integer function, is

• A. $\frac{1}{2}$
• B. $1$
• C. $2$
• D. $0$

Answer 1

The correct option is D $0$

As $0\le {\sin }^{2}x\le 1,0\le \frac{{\sin }^{2}x}{2}\le \frac{1}{2}$
$\Rightarrow \left[\frac{{\sin }^{2}x}{2}\right]=0$
Similarly, $\left[\frac{{\cos }^{2}x}{2}\right]=0$
$\therefore f\left(x\right)=\left[\frac{{\sin }^{2}x}{2}\right]+\left[\frac{{\cos }^{2}x}{2}\right]=0$