Question

The values of $x$ for which $\sqrt{\cos x-\sin x}$ is defined for $x\in [-\pi ,\pi ]$ is

• A. $\left(-\pi ,\frac{\pi }{4}\right]$
• B. $\left(0,\pi \right]$
• C. $\left(-\frac{3\pi }{4},\frac{\pi }{4}\right]$

Answer 1

The correct option is C $\left(-\frac{3\pi }{4},\frac{\pi }{4}\right]$

Given the function: $\sqrt{\cos x-\sin x}$
Now, the function is defined when $\cos x\ge \sin x$
In the given interval i.e. $x\in [-\pi ,\pi ]$
Let's take use of the graph of both the functions in the interval where $\cos x\ge \sin x$


As per the combined graph of $\sin x\&\cos x$
$\sin x\ge \cos x\forall x\in [-\pi +\frac{\pi }{4},\frac{\pi }{4}]$
$\Rightarrow \sqrt{\cos x-\sin x}$ is defined for

$x\in [-\frac{3\pi }{4},\frac{\pi }{4}]$