The Fundamental Trigonometric Identity & Reduction Formulas
1. The Unit Circle Foundation
Let's start with the unit circle — a circle of radius 1 centered at the origin. Any point on this circle can be written as:
$P = (\cos\alpha,\; \sin\alpha)$
Since the point $P(x, y)$ lies on the circle $x^2 + y^2 = 1$, we can substitute $x = \cos\alpha$ and $y = \sin\alpha$ to get the most important identity in all of trigonometry.
Interactive visualization — available in the full lesson
Drag the slider to change the angle $\alpha$. Notice that no matter what angle you choose, $\cos^2\alpha + \sin^2\alpha$ always equals 1!
2. The Fundamental Trigonometric Identity
$$\sin^2\alpha + \cos^2\alpha = 1$$
This identity holds for every angle $\alpha$. It is sometimes called the Pythagorean Identity because it comes directly from the Pythagorean theorem applied to the unit circle.
Proof:
Consider a point $P$ on the unit circle making angle $\alpha$ with the positive $x$-axis.
- The coordinates of $P$ are $(\cos\alpha,\; \sin\alpha)$.
- The distance from the origin to $P$ is the radius $r = 1$.
- By the distance formula: $\sqrt{\cos^2\alpha + \sin^2\alpha} = 1$
- Squaring both sides: $\cos^2\alpha + \sin^2\alpha = 1$ $\blacksquare$
Useful Rearrangements:
Solving for $\sin^2\alpha$:
$\sin^2\alpha = 1 - \cos^2\alpha$Solving for $\cos^2\alpha$:
$\cos^2\alpha = 1 - \sin^2\alpha$Derived Identities (dividing through):
Divide by $\cos^2\alpha$:
$\tan^2\alpha + 1 = \dfrac{1}{\cos^2\alpha}$Divide by $\sin^2\alpha$:
$1 + \cot^2\alpha = \dfrac{1}{\sin^2\alpha}$