The formula
sin²x + cos²x = 1 · tan x = sin x / cos x · sin 2x = 2 sin x cos x · cos 2x = cos²x − sin²x · 1 + tan²x = sec²x
Why it works
An identity is an equality that holds for every value of the variable, not just some: sin²x + cos²x = 1 holds for any x, whereas sin x = x holds only at zero. That is why the check cannot be done at a single point: two different expressions can agree at one by chance. The calculator evaluates both sides at eight angles spread between 0 and 2π, in radians, and skips those that make either side undefined, such as tan x at π/2. If they agree at all of them, the equality is an identity in all likelihood; if they differ at any, it is not, and that point is the counterexample. It is exactly what one does by hand with a scientific calculator before attempting the proof.
How to solve it by hand
- Write both sides of the equality with x as the variable, in radians
- Evaluate each side at several angles, skipping those that make a denominator zero
- If they differ at any, it is not an identity: that angle is the counterexample
- If they agree at all, it is an identity; to prove it, transform one side until it becomes the other
What is worth knowing
The basic identities are few and every other one follows from them. The Pythagorean one, sin²x + cos²x = 1, divided by cos²x gives 1 + tan²x = sec²x and divided by sin²x gives 1 + cot²x = csc²x. The double-angle ones come from the sum formulas with both angles equal: sin 2x = 2 sin x cos x and cos 2x = cos²x − sin²x, which with the Pythagorean identity is also written 2cos²x − 1 or 1 − 2sin²x. A very common mistake is cos 2x = 1 − sin²x, which looks like both correct forms and is neither: the calculator takes it apart at the first point. To prove an identity, start from the more complicated side and transform it, without moving terms across as if it were an equation, because that already assumes what is to be proved.