Trig Identities 31645F

1. **State the problem:** Prove the identity $$\cos^4 \alpha - \sin^4 \alpha = \cos^2 \alpha - \sin^2 \alpha$$ 2. **Use the difference of squares formula:** $$a^2 - b^2 = (a-b)(a+b)$$ 3. Apply it to the left side: $$\cos^4 \alpha - \sin^4 \alpha = (\cos^2 \alpha - \sin^2 \alpha)(\cos^2 \alpha + \sin^2 \alpha)$$ 4. Use the Pythagorean identity: $$\cos^2 \alpha + \sin^2 \alpha = 1$$ 5. Substitute back: $$= (\cos^2 \alpha - \sin^2 \alpha) \times 1 = \cos^2 \alpha - \sin^2 \alpha$$ 6. **Conclusion:** The identity holds true. --- 1. **State the problem:** Prove the identity $$\cos^2 \alpha (1 + \tan^2 \alpha) = 1$$ 2. **Recall the identity:** $$1 + \tan^2 \alpha = \sec^2 \alpha$$ 3. Substitute: $$\cos^2 \alpha (\sec^2 \alpha) = \cos^2 \alpha \times \frac{1}{\cos^2 \alpha}$$ 4. Simplify: $$= \cancel{\cos^2 \alpha} \times \frac{1}{\cancel{\cos^2 \alpha}} = 1$$ 5. **Conclusion:** The identity is proven. --- 1. **State the problem:** Prove the identity $$ (1 + \tan^2 \alpha) \times \sin^2 \alpha = \tan^2 \alpha $$ 2. Use the identity: $$1 + \tan^2 \alpha = \sec^2 \alpha$$ 3. Substitute: $$\sec^2 \alpha \times \sin^2 \alpha$$ 4. Express in terms of sine and cosine: $$\frac{1}{\cos^2 \alpha} \times \sin^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$$ 5. Recognize the right side: $$\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}$$ 6. **Conclusion:** The identity holds. --- 1. **State the problem:** Prove the identity $$\sin \alpha \cos \alpha = \frac{\tan \alpha}{1 + \tan^2 \alpha}$$ 2. Express $$\tan \alpha$$ as $$\frac{\sin \alpha}{\cos \alpha}$$: $$\frac{\tan \alpha}{1 + \tan^2 \alpha} = \frac{\frac{\sin \alpha}{\cos \alpha}}{1 + \left(\frac{\sin \alpha}{\cos \alpha}\right)^2}$$ 3. Simplify the denominator: $$1 + \frac{\sin^2 \alpha}{\cos^2 \alpha} = \frac{\cos^2 \alpha + \sin^2 \alpha}{\cos^2 \alpha} = \frac{1}{\cos^2 \alpha}$$ 4. Substitute back: $$\frac{\frac{\sin \alpha}{\cos \alpha}}{\frac{1}{\cos^2 \alpha}} = \frac{\sin \alpha}{\cos \alpha} \times \cos^2 \alpha = \sin \alpha \cos \alpha$$ 5. **Conclusion:** The identity is verified.