Question

Which of the following is/are correct regarding their fundamental period?

• A. $\frac{1}{|\sin x|+|\cos x|}\to \pi /2$
• B. $|\sin x+\cos x|\to \pi$
• C. $|\sin x+\cos x|+|\sin x-\cos x|\to 3\pi /2$
• D. $\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}\to 2\pi$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\frac{1}{|\sin x|+|\cos x|}\to \pi /2$
B $|\sin x+\cos x|\to \pi$
D $\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}\to 2\pi$

If $f\left(x\right),g\left(x\right)$ are complementary even functions, then period of $f\left(x\right)\pm g\left(x\right)$ is
$\frac{1}{2}\text{(L.C.M of period of}f\left(x\right)\text{, period of}g\left(x\right)$
and $|\sin x|,|\cos x|$ are complementary function because
$\left|\sin (\frac{\pi }{2}\pm x)\right|=|\cos x|$
and $\left|\cos (\frac{\pi }{2}\pm x)\right|=|\sin x|$
also $|\sin (-x)|=|\sin x|$ and $|\cos (-x)|=|\cos x|$
hence $|\sin x|$ and $|\cos x|$ are even function.

$\left(a\right)$ We know that the fundamental period of $|\sin x|$ and $|\cos x|$ is $\pi$
Now,
The period of
$|\sin x\mid +\mid \cos x\mid \to \frac{1}{2}(L.C.M(\pi ,\pi \left)\right)=\frac{\pi }{2}$​
So the period of
$\frac{1}{|\sin x|+|\cos x|}\to \pi /2$

$\left(b\right)|\sin x+\cos x|=\sqrt{2}|\sin (45°+x)|$
So, the period of $\sqrt{2}|\sin (45°+x)|$ is $\pi$,
Therefore, the period of
$|\sin x+\cos x|\to \pi$

$\left(c\right)$The period of $|\sin x\pm \cos x|$ is $\pi$
$\because |\sin x+\cos x\mid +\mid \sin x-\cos x|$ are complementary functions​.
$\therefore$ period of $|\sin x+\cos x\mid +\mid \sin x-\cos x\mid$
$=\frac{1}{2}(L.C.M(\pi ,\pi \left)\right)=\frac{\pi }{2}$


$\left(d\right)$ period of $\frac{|\sin x|}{\cos x}=\text{L.C.M.}(\pi ,2\pi )=2\pi$
period of $\frac{\sin x}{|\cos x|}=\text{L.C.M.}(2\pi ,\pi )=2\pi$
$\therefore$ period of $\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}=\text{L.C.M.}(2\pi ,2\pi )=2\pi$