Which Pythagorean identity relates sine and cosine?

The correct relationship between sine and cosine is expressed by the identity that states the sum of the squares of sine and cosine of an angle equals one. This foundational identity, sin²x + cos²x = 1, is pivotal in trigonometry and appears frequently in various mathematical contexts, including calculus, geometry, and algebra.

This identity implies that for any angle x, when you square the sine of that angle and add it to the square of the cosine of the same angle, the result will always be one. This reflects a fundamental relationship where sine and cosine are linked through a circular representation, such as the unit circle, where the radius is always one. Consequently, this identity can be visualized geometrically, with the x-coordinate represented by cosine and the y-coordinate represented by sine.

In contrast, other relationships offered do not accurately portray the linkage between sine and cosine, either proposing incorrect values or misrepresenting their relationship. For instance, the equation stating that their squares sum to zero does not hold true for real angles, and the assertion that their squares might equal two is not possible since both sine and cosine can only take values between -1 and 1.