How to prove the identity `sin^2x + cos^2x = 1` ?

The equation of the unit circle is x^2+y^2=1.

All points on this circle have coordinates that make this equation true.

For any random point (x, y) on the unit circle, the coordinates can be represented by (cos `theta` , sin `theta` ) where `theta` is the degrees of rotation from the positive x-axis (see attached image).

By substituting cos `theta` = x and sin `theta` = y into the equation of the unit circle, we can see that (cos `theta` )^2 + (sin `theta` )^2 = 1.

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Another method is knowing to take the derivative of

f(x) = sin^2(x) + cos^2(x)

f '(x) = 2 sin(x) cos(x) + 2 cos(x) (-sin(x))

= 2 sin(x) cos(x) - 2 cos(x) sin(x)

= 0

Since the derivative is zero everywhere the function must be a constant.

Take f(0) = sin^2(0) + cos^2(0) = 0 + 1 = 1

So

sin^2(x) + cos^2(x) = 1 everywhere.

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An alternate approach to proving this identity involves using the "unit circle" (radius = 1).  Since the radius is also the hypotenuse of the right triangle formed by the angle "x" within the circle, the sine is y and the cosine is x.  By the Pythagorean Theorem, x^2 + y^2 = 1^2 ... or x^2 + y^2 = 1