# Sum of Squares of Sine and Cosine/Proof 2

## Theorem

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

## Proof

From the trigonometric definitions of sine and cosine:

 $\displaystyle \sin x$ $=$ $\displaystyle \frac{\text{opposite} } {\text{hypotenuse} }$ $\displaystyle \cos x$ $=$ $\displaystyle \frac{\text{adjacent} } {\text{hypotenuse} }$

Then:

 $\displaystyle \sin^2 x + \cos^2 x$ $=$ $\displaystyle \frac{\text{opposite}^2 + \text{adjacent}^2} {\text{hypotenuse}^2}$ $\displaystyle$ $=$ $\displaystyle \frac{\text{hypotenuse}^2} {\text{hypotenuse}^2}$ Pythagoras's Theorem $\displaystyle$ $=$ $\displaystyle 1$

$\blacksquare$