\( f(x) = \sin^2(x)+\cos^2(x) \) Is a Constant Function

Ching-Kuang Shene (冼鏡光), Professor Emeritus
Department of Computer Science
Michigan Technological University
Houghton, MI 49931
USA
Created September 29, 2025

Define function \( f(x) \) as follows: \[ f(x) = \sin^2(x) + \cos^2(x) \] We shall prove that \( f(x) \) is a constant function.

Differentiating \( f(x) \) yields: \[ \frac{d f(x)}{dx} = \frac{d(\sin^2(x)+\cos^2(x))}{dx} = 2\sin(x)\cos(x) + 2 \cos(x)(-\sin(x)) = 0 \] Therefore, \( f(x) \) a constant function for some non-negative constant \( c \): \[ f(x) = \sin^2(x) + \cos^2(x) = c \geq 0 \] Note that because the derivatives of \( \sin(x) \) and \( \cos(x) \) can be computed without using the Pythagorean Theorem and the Pythagorean Identity (see here for the details), proving \( f(x) \) being a constant function is independent of the Pythagorean Theorem and the Pythagorean Identity.

Because \( f(x) = \sin^2(0) + \cos^2(0) = 1 \), we have the Pythagorean Identity.

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