Definition: A trigonometric identity is an equation involving trig functions that is true for every angle where both sides are defined, such as sin squared plus cos squared equals 1.
What is an identity?
Unlike an equation you solve, an identity holds for all allowed angles. The most important is the Pythagorean identity: sin2x + cos2x = 1. Dividing it by cos2x gives 1 + tan2x = sec2x.
How do you prove one?
Work on one side only, usually the more complicated one, until it matches the other. Rewrite everything in sine and cosine if you are stuck.
Worked example: prove (1 – cos2x)/sin x = sin x. Replace 1 – cos2x with sin2x, so the left side becomes sin2x / sin x = sin x.
Numeric check at x = 30 degrees: (1 – 0.75)/0.5 = 0.5, which equals sin 30.
What mistake should you avoid?
Moving terms across the equals sign, which assumes the identity is already true. Mathbot reminds you to transform one side at a time. Also, sin2x means (sin x)2, not sin(x2).