Trig Identity Cbdde9

1. **State the problem:** Prove the identity $$(\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \times \cos \theta$$. 2. **Recall the formula:** The square of a sum is given by $$(a+b)^2 = a^2 + 2ab + b^2$$. 3. **Apply the formula:** Let $a = \sin \theta$ and $b = \cos \theta$, then $$ (\sin \theta + \cos \theta)^2 = (\sin \theta)^2 + 2 \sin \theta \cos \theta + (\cos \theta)^2 $$ 4. **Use the Pythagorean identity:** We know that $$ \sin^2 \theta + \cos^2 \theta = 1 $$ 5. **Substitute:** Replace $\sin^2 \theta + \cos^2 \theta$ with 1 in the expression: $$ (\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \cos \theta $$ 6. **Conclusion:** The left side equals the right side, so the identity is proven. **Final answer:** $$(\sin \theta + \cos \theta)^2 = 1 + 2 \sin \theta \times \cos \theta$$ is an identity.