1. **Problem:** Prove the identity $$\tan \theta + \cot \theta = \sec \theta \csc \theta$$. 2. **Formula and rules:** Recall the definitions: - $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ - $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$ - $$\sec \theta = \frac{1}{\cos \theta}$$ - $$\csc \theta = \frac{1}{\sin \theta}$$ 3. **Start with the left-hand side (LHS):** $$\tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}$$ 4. **Find common denominator and combine:** $$= \frac{\sin^2 \theta}{\cos \theta \sin \theta} + \frac{\cos^2 \theta}{\cos \theta \sin \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta}$$ 5. **Use Pythagorean identity:** $$\sin^2 \theta + \cos^2 \theta = 1$$ 6. **Simplify numerator:** $$= \frac{1}{\cos \theta \sin \theta}$$ 7. **Rewrite denominator using sec and csc:** $$= \frac{1}{\cos \theta} \times \frac{1}{\sin \theta} = \sec \theta \csc \theta$$ 8. **Conclusion:** LHS equals RHS, so the identity is proven. **Final answer:** $$\tan \theta + \cot \theta = \sec \theta \csc \theta$$