What is a key identity in trigonometry?

The key identity in trigonometry referenced in this question is sin²x + cos²x = 1. This identity is fundamental because it establishes a relationship between the sine and cosine of an angle, which are the primary functions in trigonometry. This equation holds true for all angles, providing a basis for various calculations and proofs in trigonometric analysis.

This identity is crucial in the unit circle context, where the sine of an angle corresponds to the y-coordinate and the cosine corresponds to the x-coordinate of a point on the circle. Since any point on the unit circle must satisfy the equation x² + y² = 1, replacing x with cos(x) and y with sin(x) confirms the integrity of this identity.

The other options do not present valid or commonly used trigonometric identities. For example, tan x = cos x/sin x is incorrect since the tangent function is defined as sin x/cos x, while sin²x + cos²x = 0 contradicts the Pythagorean theorem as sine and cosine values range between -1 and 1, making their squares sum to 1, not 0. Finally, the expression tan x = sin x + cos x does not represent a