A trigonometric identity is a mathematical equation that involves trigonometric functions (such as sine, cosine, and tangent) and is true for all values of the variables involved.
To show that a trigonometric identity adds up to 0, you need to manipulate the equation using algebraic and trigonometric identities until it simplifies to 0. This can involve using properties such as the sum and difference identities, double angle identities, and Pythagorean identities.
Understanding trigonometric identities is important because they are used in many areas of mathematics and science, such as calculus, physics, and engineering. They also provide a way to simplify complex trigonometric expressions and solve equations.
One example is the identity sin^2(x) + cos^2(x) = 1. By subtracting cos^2(x) from both sides, we get sin^2(x) = 1 - cos^2(x), which simplifies to sin^2(x) + cos^2(x) - 1 = 0.
Trigonometric identities are used in many real-life applications, such as measuring distances and angles in navigation and surveying, analyzing sound waves and vibrations, and designing structures and machines. Understanding these identities allows us to accurately model and solve these real-world problems.