Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the identity are defined. These identities are fundamental in mathematics, especially in fields such as geometry, calculus, and physics.
One common example of a trigonometric identity is the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
This identity states that for any angle θ, the square of the sine of that angle plus the square of the cosine of that angle equals one. It can be derived from the Pythagorean theorem in a right triangle, where the hypotenuse is the radius of a unit circle, and the opposite and adjacent sides represent the sine and cosine of the angle, respectively.
These identities are incredibly useful for simplifying expressions, solving equations, and proving other mathematical concepts.