Question

Which of the following is TRUE ?

• A. The fundamental period of $|\sin x|+|\cos x|$ is $\pi$
• B. The fundamental period of $|\sin x\cdot \cos x|$ is $\pi$.
• C. The fundamental period of $|\sin x|-|\cos x|$ is $\pi$
• D. The fundamental period of $|\sin x|$ is $2\pi$

Answer 1

The correct option is C The fundamental period of $|\sin x|-|\cos x|$ is $\pi$

The fundamental period of $|\sin x|$ is $\pi$

Let $f\left(x\right)=|\sin x|+|\cos x|$
$f(\frac{\pi }{2}+x)=|-\sin x|+|\cos x|=|\sin x|+|\cos x|=f\left(x\right)$
Therefore, period is $\frac{\pi }{2}$

Let $f\left(x\right)=|\sin x|-|\cos x|$
$f(\frac{\pi }{2}+x)=|\cos x|-|-\sin x|=-f\left(x\right)f(\pi +x)=|\sin x|-|\cos x|=f\left(x\right)$
$\Rightarrow$ Period is $\pi$

Let $f\left(x\right)=|\sin x\cdot \cos x|=\frac{1}{2}|\sin 2x|$
Therefore, period is $\frac{\pi }{2}$