Proving the Pythagorean Identity: A Fundamental Trigonometry Identity Explained

sin^2x + cos^2x =

The equation sin^2x + cos^2x = 1 is one of the fundamental identities in trigonometry known as the Pythagorean identity

The equation sin^2x + cos^2x = 1 is one of the fundamental identities in trigonometry known as the Pythagorean identity.

To prove this identity, we’ll start with the Pythagorean theorem from geometry, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Next, consider a unit circle with its center at the origin (0,0) in the coordinate plane. The radius of this unit circle is 1 unit. Now, draw a line from the center of the circle to any point on its circumference, creating an angle with respect to the positive x-axis. This line is the radius of the circle, and the angle it forms is the angle x.

Let’s call the x-coordinate of the point on the circumference as cos(x) and the y-coordinate as sin(x). From the Pythagorean theorem, we have:

cos^2x + sin^2x = r^2

Since we are dealing with a unit circle, the radius is 1, so r^2 = 1. Substituting this back into the equation, we get:

cos^2x + sin^2x = 1

Therefore, sin^2x + cos^2x = 1, which is the Pythagorean identity.

This identity is extremely useful in trigonometry as it allows us to relate sine and cosine functions to each other and simplifies calculations involving these functions.

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