Proving the Pythagorean Identity: Sin²x + Cos²x = 1 Using a Unit Circle and Trigonometric Principles

sin²x + cos²x = 1

To prove the equation sin²x + cos²x = 1, we will make use of a fundamental trigonometric identity

To prove the equation sin²x + cos²x = 1, we will make use of a fundamental trigonometric identity.

The Pythagorean identity states that for any angle x in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two other sides. In trigonometric terms, this identity can be written as:

sin²x + cos²x = 1

To understand this equation, we can consider a unit circle. A unit circle has a radius of 1 and is centered at the origin (0, 0) in a Cartesian coordinate system.

Let’s consider a point P on the unit circle. The x-coordinate of point P is cos(x) and the y-coordinate is sin(x). Since the radius is 1, the length of the hypotenuse can be calculated using the Pythagorean theorem:

Length of hypotenuse = √(cos²x + sin²x)

Since we are working with a unit circle, the length of the hypotenuse is equal to 1:

√(cos²x + sin²x) = 1

Squaring both sides of the equation, we get:

cos²x + sin²x = 1

Therefore, we have proven that sin²x + cos²x = 1 using the Pythagorean identity.

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