<s>[INST] Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$
I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine.
 [/INST] Consider a right-angled triangle, $\Delta ABC$, where $\angle BAC = \theta$,

By the Pythagorean theorem,
$$ {AC}^2+{BC}^2 = {AB}^2 $$
Dividing by $AB^2$,
$$
\require{cancel}
\begin{align}
&\Rightarrow \frac{AC^2}{AB^2} + \frac{BC^2}{AB^2} = \frac{AB^2}{AB^2}\\
&\Rightarrow \Big(\frac{\text{opposite}}{\text{hypotenuse}}\Big)^2 + \Big(\frac{\text{adjacent}}{\text{hypotenuse}}\Big)^2 = \frac{\cancel{AB^2}}{\cancel{AB^2}} = 1\\
&\Rightarrow \boxed{\sin^2\theta + \cos^2\theta = 1}
\end{align}
$$
</s>