1. **State the problem:** Verify the trigonometric identity $$\cot \beta + \tan \beta = \sec \beta \csc \beta$$. 2. **Recall definitions and formulas:** - $$\cot \beta = \frac{\cos \beta}{\sin \beta}$$ - $$\tan \beta = \frac{\sin \beta}{\cos \beta}$$ - $$\sec \beta = \frac{1}{\cos \beta}$$ - $$\csc \beta = \frac{1}{\sin \beta}$$ 3. **Start with the left-hand side (LHS):** $$\cot \beta + \tan \beta = \frac{\cos \beta}{\sin \beta} + \frac{\sin \beta}{\cos \beta}$$ 4. **Find common denominator and combine:** $$= \frac{\cos^2 \beta}{\sin \beta \cos \beta} + \frac{\sin^2 \beta}{\sin \beta \cos \beta} = \frac{\cos^2 \beta + \sin^2 \beta}{\sin \beta \cos \beta}$$ 5. **Use Pythagorean identity:** $$\cos^2 \beta + \sin^2 \beta = 1$$ 6. **Simplify numerator:** $$= \frac{1}{\sin \beta \cos \beta}$$ 7. **Rewrite denominator using sec and csc:** $$= \frac{1}{\sin \beta} \cdot \frac{1}{\cos \beta} = \csc \beta \sec \beta$$ 8. **Conclusion:** LHS equals RHS, so the identity is verified: $$\cot \beta + \tan \beta = \sec \beta \csc \beta$$