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Here, $a=1$, $b=1$, and $c=1$.

The sum of the roots is given by the formula: $\alpha + \beta = -\frac{b}{a}$

The product of the roots is given by the formula: $\alpha \beta = \frac{c}{a}$

First, simplify the expression:

Now, substitute the values obtained from Step 2 and Step 3:

Therefore, the value of $\alpha + \beta$ is $-1$ and the value of $\frac{1}{\alpha} + \frac{1}{\beta}$ is $-1$.

$\boxed{\alpha + \beta = -1, \quad \frac{1}{\alpha} + \frac{1}{\beta} = -1}$