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1 is the identity element of multiplication.
If you add the identity element - namely zero - to a number, you will get the same number back.
Just as with any other identity, a trigonometric identity is a trigonometric statement (other than a definition), which is true for all values of the variable or variables.
x times 1 = x illustrates the multiplicative identity property.
x=y is the identity. It is its own inverse. So the inverse is y=x.