Prove that sin^2(x) + cos^2(x) = 1

sin^2(x) + cos^2(x) = 1 proof

Show that

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

.

Proof. Recall the next addition formula for Cosine:

\begin{align*} \cos(A-B) = \cos(A)\cos(B) + \sin(A)\sin(B). \end{align*}

Now take

A = B = x

. Then we get:

\begin{align*} \cos(x - x) = \cos(x)\cos(x) + \sin(x)\sin(x) &\iff \cos(0) = \cos^2(x) + \sin^2(x) \\ &\iff 1 = \cos^2(x) + \sin^2(x). \end{align*}

So we get

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