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.
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)
Since the derivative is zero everywhere the function must be a constant.
Take f(0) = sin^2(0) + cos^2(0) = 0 + 1 = 1
sin^2(x) + cos^2(x) = 1 everywhere.
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