On the schröter formula for theta functions

The Schröter formula is an important theta function identity. In this paper, we will point out that some well-known addition formulas for theta functions are special cases of the Schröter formula. We further show that the Hirschhorn septuple product identity can also be derived from this formula. In addition, this formula allows us to derive four remarkable theta functions identities, two of them are extensions of two well-known Ramanujan's identities related to the modular equations of degree 5. A trigonometric identity is also proved.