cos²(x) + sin²(x) = 1: Pythagoras in disguise

The identity

For any angle $x$ (in degrees or radians):

$\cos^2(x) + \sin^2(x) = 1$

Why it’s true: Pythagoras on the unit circle

A point on the circle of radius $1$ at angle $x$ has coordinates $(\cos x, \sin x)$. The right triangle formed by this point, the origin, and its projection on the x-axis has sides $\cos x$, $\sin x$ and hypotenuse $1$.

Pythagoras: $(\cos x)^2 + (\sin x)^2 = 1^2 = 1$.

What it’s for (in practice)

• Know $\cos x$, compute $\sin x$ (up to sign): $\sin x = \pm\sqrt{1 - \cos^2 x}$.
• Simplify trig expressions: whenever you see $\cos^2 + \sin^2$, replace it with 1.
• Prove other identities ($\tan$ formula, expansions).

The exam use

Many problems ask you to « show that » something equals 1 or 0 — the key move is almost always this identity.

?Your turn

If sin(x) = 3/5, what is cos²(x)?

A.9/25B.16/25C.4/5D.1 − 3/5

Check