How can we try to grep and egrep on two words i.e. both words should exists in the line.

something like

ta0kira

edit:
If you need a single statement, try this.

I mean by using single egrep... looks like logical OR is working , but not logical AND.

Similarly also looking for grep command as well

Please see my edit.

grep uses regexes, not logical statements. Regexes give you some logical options, but those options are to express alternatives in a pattern. "And" is implied unless "|" is used, but what you're talking about is parallel analysis rather than a single expression, though it can be made to work in this simple example.
ta0kira

PS Note that "|" itself is "xor", meaning that a position is filled by one or the other, but not both. If you use "*" or "+", you just denote repetition of the previous statement, which is entirely arbitrary.

Try the -P (treat as perl regex) option

ls | grep -P word1.*word2