The identity sin²θ + cos²θ equals:

### The Pythagorean Identity The expression $\sin^2\theta + \cos^2\theta$ is known as the **fundamental Pythagorean identity**. **1. Geometric Derivation** Consider a right-angled triangle with an angle $\theta$, an adjacent side $b$, an opposite side $a$, and a hypotenuse $c$. By definition: $$\sin \theta = \frac{a}{c} \quad \text{and} \quad \cos \theta = \frac{b}{c}$$ **2. Substitution** If we square both terms and add them together: $$\sin^2 \theta + \cos^2 \theta = \left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = \frac{a^2}{c^2} + \frac{b^2}{c^2} = \frac{a^2 + b^2}{c^2}$$ **3. Applying the Pythagorean Theorem** According to the Pythagorean theorem, for any right triangle, $a^2 + b^2 = c^2$. Substituting this into our fraction: $$\frac{a^2 + b^2}{c^2} = \frac{c^2}{c^2} = 1$$ Therefore, $\sin^2\theta + \cos^2\theta = 1$ for any value of $\theta$.