Exploring the Pythagorean Identity | cos²x + sin²x = 1

cos²x+sin²x

The expression you provided, cos²x + sin²x, is known as one of the fundamental trigonometric identities—the Pythagorean Identity

The expression you provided, cos²x + sin²x, is known as one of the fundamental trigonometric identities—the Pythagorean Identity.

In trigonometry, the Pythagorean Identity states that the square of the cosine of an angle added to the square of the sine of the same angle is always equal to 1. Symbolically, it is expressed as:

cos²x + sin²x = 1

This identity holds true for any angle x. It can be proven by using the definitions of cosine and sine in terms of a right triangle.

Consider a right triangle with angle x as one of its acute angles. Let’s assume that the length of the base of the triangle is represented by “adjacent” and the length of the side opposite the angle x is represented by “opposite.” The hypotenuse of the triangle is represented by “hypotenuse.”

Since cosine is defined as the ratio of the adjacent side to the hypotenuse, we have:

cos(x) = adjacent / hypotenuse

And, sine is defined as the ratio of the opposite side to the hypotenuse, so we have:

sin(x) = opposite / hypotenuse

Using the Pythagorean Theorem, we know that the sum of the squares of the lengths of the two shorter sides (adjacent and opposite) is equal to the square of the length of the longest side (hypotenuse):

adjacent² + opposite² = hypotenuse²

Dividing both sides of this equation by the square of the hypotenuse, we get:

(adjacent² / hypotenuse²) + (opposite² / hypotenuse²) = (hypotenuse² / hypotenuse²)

This simplifies to:

(adjacent / hypotenuse)² + (opposite / hypotenuse)² = 1

Substituting the definitions of cosine and sine from above, we have:

cos²x + sin²x = 1

Therefore, the expression cos²x + sin²x is always equal to 1, irrespective of the value of angle x.

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