Trig Identities Fc8D5C

1. **State the problem:** Use the given trigonometric identities and reduction formulas to solve problems. 2. **Recall the identities:** - $\tan\theta = \frac{\sin\theta}{\cos\theta}$ - $\sin^2 \theta + \cos^2 \theta = 1$ - $1 + \tan^2 \theta = \sec^2 \theta$ - $1 + \cot^2 \theta = \csc^2 \theta$ 3. **Reduction formulas for angles:** - $\sin(180^\circ - \theta) = \sin\theta$ - $\cos(180^\circ - \theta) = -\cos\theta$ - $\tan(180^\circ - \theta) = -\tan\theta$ - $\sin(360^\circ - \theta) = -\sin\theta$ - $\cos(360^\circ - \theta) = \cos\theta$ - $\tan(360^\circ - \theta) = -\tan\theta$ 4. **Problem 1:** Simplify $\tan(180^\circ - \theta)$ using identities. Using the reduction formula: $$\tan(180^\circ - \theta) = -\tan\theta$$ Using the identity $\tan\theta = \frac{\sin\theta}{\cos\theta}$, this becomes: $$\tan(180^\circ - \theta) = -\frac{\sin\theta}{\cos\theta}$$ 5. **Problem 2:** Verify $\sin^2(180^\circ - \theta) + \cos^2(180^\circ - \theta) = 1$ Using reduction formulas: $$\sin(180^\circ - \theta) = \sin\theta$$ $$\cos(180^\circ - \theta) = -\cos\theta$$ Then: $$\sin^2(180^\circ - \theta) + \cos^2(180^\circ - \theta) = \sin^2\theta + (-\cos\theta)^2 = \sin^2\theta + \cos^2\theta = 1$$ 6. **Problem 3:** Simplify $1 + \tan^2(360^\circ - \theta)$ Using reduction formula: $$\tan(360^\circ - \theta) = -\tan\theta$$ Then: $$1 + \tan^2(360^\circ - \theta) = 1 + (-\tan\theta)^2 = 1 + \tan^2\theta = \sec^2\theta$$ 7. **Summary:** These problems show how to apply the fundamental identities and reduction formulas for angles $180^\circ$ and $360^\circ$ to simplify and verify trigonometric expressions. **Final answers:** - $\tan(180^\circ - \theta) = -\tan\theta$ - $\sin^2(180^\circ - \theta) + \cos^2(180^\circ - \theta) = 1$ - $1 + \tan^2(360^\circ - \theta) = \sec^2\theta$