1. **State the problem:** Simplify the expression $$\frac{\cos^2 \theta}{1 - \tan^2 \theta} - \frac{\sin^2 \theta}{1 - \cot^2 \theta}$$. 2. **Recall identities:** - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ - $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$ - Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$ 3. **Rewrite denominators:** - $$1 - \tan^2 \theta = 1 - \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta}$$ - $$1 - \cot^2 \theta = 1 - \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta - \cos^2 \theta}{\sin^2 \theta}$$ 4. **Rewrite each fraction:** - $$\frac{\cos^2 \theta}{1 - \tan^2 \theta} = \frac{\cos^2 \theta}{\frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta}} = \frac{\cos^2 \theta \cdot \cos^2 \theta}{\cos^2 \theta - \sin^2 \theta} = \frac{\cos^4 \theta}{\cos^2 \theta - \sin^2 \theta}$$ - $$- \frac{\sin^2 \theta}{1 - \cot^2 \theta} = - \frac{\sin^2 \theta}{\frac{\sin^2 \theta - \cos^2 \theta}{\sin^2 \theta}} = - \frac{\sin^2 \theta \cdot \sin^2 \theta}{\sin^2 \theta - \cos^2 \theta} = - \frac{\sin^4 \theta}{\sin^2 \theta - \cos^2 \theta}$$ 5. **Note that $$\sin^2 \theta - \cos^2 \theta = - (\cos^2 \theta - \sin^2 \theta)$$, so rewrite the second term denominator:** - $$- \frac{\sin^4 \theta}{\sin^2 \theta - \cos^2 \theta} = - \frac{\sin^4 \theta}{- (\cos^2 \theta - \sin^2 \theta)} = \frac{\sin^4 \theta}{\cos^2 \theta - \sin^2 \theta}$$ 6. **Combine the two terms over common denominator:** - $$\frac{\cos^4 \theta}{\cos^2 \theta - \sin^2 \theta} + \frac{\sin^4 \theta}{\cos^2 \theta - \sin^2 \theta} = \frac{\cos^4 \theta + \sin^4 \theta}{\cos^2 \theta - \sin^2 \theta}$$ 7. **Simplify numerator:** - Recall $$a^4 + b^4 = (a^2 + b^2)^2 - 2a^2 b^2$$ - So $$\cos^4 \theta + \sin^4 \theta = (\cos^2 \theta + \sin^2 \theta)^2 - 2 \cos^2 \theta \sin^2 \theta = 1^2 - 2 \cos^2 \theta \sin^2 \theta = 1 - 2 \cos^2 \theta \sin^2 \theta$$ 8. **Final expression:** $$\frac{1 - 2 \cos^2 \theta \sin^2 \theta}{\cos^2 \theta - \sin^2 \theta}$$ This is the simplified form of the original expression.