Question

The general solution of $\left|\sin x=\cos x\right|$ is (when $n\in \mathrm{\mathbb{Z}}$) given by

• A. $n\pi +\frac{\pi }{4}$
• B. $2n\pi \pm \frac{\pi }{4}$
• C. $n\pi \pm \frac{\pi }{4}$
• D. $n\pi -\frac{\pi }{4}$

Answer 1

The correct option is B

$2n\pi \pm \frac{\pi }{4}$

Find the general solution

Given, $\left|\sin x=\cos x\right|$

We know that, $\left|x\right|=x$ when $x\ge 0$ and $\left|x\right|=-x$ when $x<0$

Similarly,

$\left|\sin x\right|=\sin x$ when $x\in \left[0,\pi \right]$ and $\left|\sin x\right|=-\sin x$ when $x\in \left[\pi ,2\pi \right]$

Now,$x\in \left[0,\pi \right]$

$\sin x=\cos x$

$\Rightarrow \tan x=1$

$\Rightarrow \tan x=\tan \frac{\pi }{4}$

$\Rightarrow x=n\pi +\frac{\pi }{4},n\in Z$

When $x\in (\pi ,2\pi )$

$-\sin x=\cos x$

$\Rightarrow -\tan x=1$

$\Rightarrow \tan x=-1$

$$\begin{gathered} \Rightarrow \tan x=-\tan \frac{\pi }{4} \\ \Rightarrow \tan x=\tan \left(\pi –\frac{\pi }{4}\right) \\ \Rightarrow x=n\pi +\frac{3\pi }{4},n\in Z \end{gathered}$$

Combining both the equations, we get

The general solution is $x=2n\pi \pm \frac{\pi }{4}$

Hence, option (B) is the correct answer.