sin^2 x + cos^2 x
The expression “sin^2 x + cos^2 x” represents a fundamental identity in trigonometry known as the Pythagorean identity
The expression “sin^2 x + cos^2 x” represents a fundamental identity in trigonometry known as the Pythagorean identity. It states that the sum of the squares of the sine and cosine of an angle x is always equal to 1.
To understand this identity, let’s start by defining the sine and cosine functions. In a right triangle, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
Now, consider a right triangle where the hypotenuse has a length of 1 unit. Let one of the acute angles be x. The side opposite angle x is represented by sin(x), and the side adjacent to angle x is represented by cos(x). By definition, sin(x)^2 + cos(x)^2 is the sum of the squares of these two sides.
Now, using the Pythagorean theorem, we know that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. Therefore, sin(x)^2 + cos(x)^2 = 1^2, which simplifies to sin(x)^2 + cos(x)^2 = 1.
This fundamental identity, sin^2 x + cos^2 x = 1, holds true for any angle x, and is a fundamental concept in trigonometry. It is often used to simplify trigonometric expressions and solve various trigonometric equations.
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