Euler Identity , useless formula

The quote by physicist Wolfgang Pauli mocking a certain research paper—"This isn't right. It's not even wrong!"—I find it applies to Euler's identity. Euler's identity is a completely useless relation. It offers nothing more than an alternative notation for complex numbers in certain cases. For example, it cannot be used to calculate the quantity pi from the number e; these two numbers remain independent, and each must be computed using its own separate method. Despite this, there is an excessive amount of praise and poetic glorification of this formula, mainly because it links the four most important mathematical constants—an extremely weak connection, as I will explain. Euler's identity is not a general algebraic relation linking these constants, but rather a geometric relation for a very specific case—namely when the number x represents an angle measured in radians. This is because we use the series expansion of trigonometric functions in its derivation, as shown in (2a) and (3a) in the image. This expansion is only valid when the angle is measured in radians. If we try to apply the expansion in the general case, we get the relations (2b) and (3b) instead, which do not lead to Euler's identity. While the exponential function’s series expansion depends only on the value of x as a pure number, the expansions of the trigonometric functions depend on x as an angle and on the unit in which it is measured—and only lead to Euler's identity in a specific case.