the Pythagorean identity
Among the dozens of trigonometric identities, one is the parent of nearly all the rest. It says that for every angle theta whatsoever, sin^2 theta + cos^2 theta = 1. Simple to state and impossible to break, it is the single most useful relationship in trigonometry.
Why it is true is the unit circle in one line: the point (cos theta, sin theta) sits on a circle of radius 1, and the equation of that circle is x^2 + y^2 = 1, so cos^2 theta + sin^2 theta = 1 automatically. (Here sin^2 theta is shorthand for (sin theta)^2.) For acute angles it is literally the Pythagorean theorem applied to the right triangle inside the circle, whose legs are sin theta and cos theta and whose hypotenuse is 1 — hence the name. From this one identity you can always trade a sine for a cosine: sin theta = sqrt(1 - cos^2 theta), up to sign.
Two relatives follow by dividing the identity through. Divide every term by cos^2 theta to get 1 + tan^2 theta = sec^2 theta; divide by sin^2 theta to get 1 + cot^2 theta = csc^2 theta. These three are the engine behind simplifying expressions, proving other identities, and the trigonometric substitutions of calculus. The classic beginner error is dropping the squares or the sign — sin theta + cos theta does NOT equal 1, only the sum of their squares does.
If cos theta = 3/5 and theta is acute, find sin theta. From sin^2 theta = 1 - cos^2 theta = 1 - 9/25 = 16/25, so sin theta = 4/5 (positive because theta is acute). This is exactly the 3-4-5 triangle wearing its trigonometric clothes.
Knowing one of sine or cosine instantly gives the other, up to a sign set by the quadrant.
It is the squares that add to 1, not the functions themselves; and when you take the square root to recover sin theta from cos theta, you must restore the correct sign from the angle's quadrant.