Sine squared plus cosine squared is always one

Statement

Sine squared plus cosine squared is always one:

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

Proof

Construct a right triangle with one angle being $\theta$, like the image below:

From the definitions from trigonometry follow:

$$\sin(\theta) = \frac ab \implies a = b\sin(\theta)$$

$$\cos(\theta) = \frac cb \implies c = b\cos(\theta)$$

Using the Pythagorean theorem, find that:

$$a^2 + c^2 = b^2$$

When substituting $a$ and $c$, you get:

$$\bigg(b\sin(\theta)\bigg)^2 + \bigg(b\cos(\theta)\bigg)^2 = b^2$$

$$b^2 \sin(\theta)^2 + b^2 \cos(\theta)^2 = b^2$$

Finally, divide each term by $b^2$.

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