sin^2x + cos^2x =
The equation sin^2x + cos^2x = 1 is a fundamental trigonometric identity known as the Pythagorean identity
The equation sin^2x + cos^2x = 1 is a fundamental trigonometric identity known as the Pythagorean identity. It states that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1.
This identity arises from the properties of right triangles and the unit circle. When you draw a unit circle (a circle with a radius of 1) centered at the origin on a coordinate plane, the sine of an angle is defined as the y-coordinate of the point where a terminal side of the angle intersects the unit circle, and the cosine of the angle is defined as the x-coordinate of the same point.
In a right triangle, the lengths of the two shorter sides squared, added together, will equal the length of the longest side (the hypotenuse) squared. Applying this concept to the unit circle, the x-coordinate squared plus the y-coordinate squared will always equal 1 squared, which is 1.
Using trigonometric ratios, we can express sin^2x as (1 – cos^2x) and cos^2x as (1 – sin^2x). Substituting these expressions into the Pythagorean identity, we get (1 – cos^2x) + cos^2x = 1. Simplifying the equation gives 1 – cos^2x + cos^2x = 1, and the cos^2x terms cancel each other out, leaving us with 1 = 1.
Therefore, sin^2x + cos^2x will always result in 1 due to the Pythagorean identity. This identity is widely used in various areas of trigonometry and calculus to simplify expressions and solve equations involving sine and cosine functions.
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