Match the following

	List I		List II
1	Period of $\sin \left(e^{\tan x}+e^{\cot x}\right)$	a	does not exist
2	Period of $\cos \left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\dots .\mathrm{\infty }\right)x$	b	$\mathrm{\pi }$
3	Period of$\sin x^2$	c	$\frac{\mathrm{\pi }}{3}$
4	Period of $\sin (x+2x+3x)$	d	$2\mathrm{\pi }$

Period of tanx,cotx  is $\mathrm{\pi }\Rightarrow$ Period of  $\sin \left(e^{\tan x}+e^{\cot x}\right)$ is $\mathrm{\pi }$

$$\begin{gathered} \cos \left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\dots .\mathrm{\infty }\right)x=\cos \left(\frac{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right)=\cos x \\ \therefore period =2\mathrm{\pi } \end{gathered}$$

$\sin x^2$ is non periodic So period does not exist

$$\begin{gathered} \sin (x+2x+3x)=\sin 6x \\ \Rightarrow \mathrm{period} \mathrm{is}\frac{2\mathrm{\pi }}{6}=\frac{\mathrm{\pi }}{3} \end{gathered}$$