1. **State the problem:** Simplify the expression $$\frac{1}{\cot x + 1} + \frac{1}{\tan x + 1}$$. 2. **Recall the definitions:** - $$\tan x = \frac{\sin x}{\cos x}$$ - $$\cot x = \frac{\cos x}{\sin x}$$ 3. **Rewrite the expression using these definitions:** $$\frac{1}{\frac{\cos x}{\sin x} + 1} + \frac{1}{\frac{\sin x}{\cos x} + 1}$$ 4. **Find common denominators inside each denominator:** $$\frac{1}{\frac{\cos x + \sin x}{\sin x}} + \frac{1}{\frac{\sin x + \cos x}{\cos x}}$$ 5. **Invert and multiply:** $$\frac{\sin x}{\cos x + \sin x} + \frac{\cos x}{\sin x + \cos x}$$ 6. **Since denominators are the same, combine:** $$\frac{\sin x + \cos x}{\sin x + \cos x}$$ 7. **Simplify:** $$1$$ **Final answer:** $$1$$