Question

The domain of the function $f\left(x\right)=\sqrt{{\log }_{\sin x+\cos x}(\left|\cos x\right|+\cos x)},0\le x\le \pi$ is

• A. $(0,\pi )$
• B. $(0,\frac{\pi }{2})$
• C. $(0,\frac{\pi }{3})$
• D. None of these

Answer 1

The correct option is C $(0,\frac{\pi }{3})$

$\{\begin{array}{c}|\cos x|+\cos x>0;x\in [0,\frac{\pi }{2})\\ |\cos x|+\cos x=0;x\in [\frac{\pi }{2},\pi ]\end{array}$

For $x\in [0,\frac{\pi }{2})$,

$1<\sin x+\cos x\le \sqrt{2}$

Now, for $x\in (0,\frac{\pi }{2})$

${\log }_{\sin x+\cos x}(\left|\cos x\right|+\cos x)\ge 0$

As for ${\log }_{a}x$, $a>1$, so $x$ must be $\ge 0$

$|\cos x|+\cos x\ge 0$

As we are considering $x\in (0,\frac{\pi }{2})$ ... (1)

$\Rightarrow \cos x+\cos x\ge 1$

$\Rightarrow 2\cos x\ge 1$

$\Rightarrow \cos x\ge \frac{1}{2}$

Using (1), we can say that

$x\in (0,\frac{\pi }{3})$

Hence, option C.