$$\frac{1}{1+2^{a-b}}+\frac{1}{1+2^{b-a}}$$ is
Expression : $$\frac{1}{1+2^{a-b}}+\frac{1}{1+2^{b-a}}$$
= $$\frac{1}{1 + \frac{2^a}{2^b}} + \frac{1}{\frac{2^b}{2^a}}$$
= $$\frac{2^b}{2^a + 2^b} + \frac{2^a}{2^a + 2^b}$$
= $$\frac{2^a + 2^b}{2^a + 2^b} = 1$$
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