The Pythagorean Identity (F 706)

Picture a point spinning around a circle of radius 1, centered at the origin. At any angle x, that point sits at coordinates (cos(x), sin(x)). Draw a right triangle from the center to that point: the horizontal leg has length cos(x), the vertical leg has length sin(x), and the hypotenuse is the radius, length 1. The Pythagorean theorem says leg² + leg² = hypotenuse², so cos²(x) + sin²(x) = 1² = 1. That's the whole identity — the circle's radius being 1 is what makes the right side exactly 1, for every angle, forever.

This identity is the main tool for moving between sine and cosine when you only know one of them. Given sin(x), solve cos²(x) = 1 - sin²(x), then take a square root — and use the quadrant to decide if cos(x) is positive or negative, since the square root alone can't tell you the sign. It also lets you rewrite any expression mixing sin²(x) and cos²(x) in terms of just one function, which is often the first move in simplifying a trig expression or proving another identity.

Divide every term by cos²(x) and you get tan²(x) + 1 = sec²(x), a second identity that connects to the tangent and secant graphs. Divide by sin²(x) instead and you get 1 + cot²(x) = csc²(x). All three come from the same triangle on the unit circle, just scaled differently.