Does the president of the united states have the power to declare war without the approval of the congress by issuing an executive order?

No he cannot, it has to be passed through congress.

Though we have not declared war formally since 1942. We have used military without the declaration of war.

the triple alliance. germany and austria-hungary were already allies by 1879 when, worried about a threat from russia, they formed the dual alliance. italy joined the group in 1882. there was a concern about a possible attack from france and the alliance needed powerful friends to defend it.