SL 3.6—Pythagorean identity, double angles

One of the most fundamental identities in trigonometry comes directly from the Pythagorean theorem. Consider a right-angled triangle with opposite side $o$, adjacent side $a$, and hypotenuse $h$. By the Pythagorean theorem:

$o^2+a^2=h^2$

Dividing every term by $h^2$:

$\frac{o^2}{h^2}+\frac{a^2}{h^2}=1\Rightarrow {\left(\frac{o}{h}\right)}^2+{\left(\frac{a}{h}\right)}^2=1$

Since $\sin \theta =\frac{o}{h}$ and $\cos \theta =\frac{a}{h}$, this becomes:

${\cos }^2\theta +{\sin }^2\theta =1$